Ehlers, John F. - Cybernetic Analysis for Stocks and Futures [Part 1]

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CyberneticAnalysis forStocks and

Futures Cutting-Edge DSP Technology

to Improve Your Trading

JOHN F. EHLERS

John Wiley & Sons, Inc.

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Futures

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Founded in 1807, John Wiley & Sons is the oldest independent publish-ing company in the United States. With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding.

The Wiley Trading series features books by traders who have survived the market’s ever changing temperament and have prospered—some by rein-venting systems, others by getting back to basics. Whether a novice trader, professional, or somewhere in between, these books will provide the advice and strategies needed to prosper today and well into the future.

For a list of available titles, visit our Web site at www.WileyFinance.com.

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Futures Cutting-Edge DSP Technology

to Improve Your Trading

JOHN F. EHLERS

John Wiley & Sons, Inc.

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Copyright © 2004 by John F. Ehlers. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or oth-erwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through pay-ment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright .com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies con-tained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Some of the charts in this book were created using TradeStation, copyright © TradeStation Securities, Inc., 2000–2004.

TradeStation and EasyLanguage are registered trademarks of TradeStation Technologies, Inc., an affiliate of TradeStation Securities.

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Library of Congress Cataloging-in-Publication Data:

Ehlers, John F., 1933-Cybernetic analysis for stocks and futures : cutting-edge DSP

technology to improve your trading / John F. Ehlers. p. cm.

Includes bibliographical references. ISBN 0-471-46307-8 1. Corporations—Valuation. 2. Chief executive officers—Rating of.

3. Investment analysis. I. Title.HG4028.V3 E365 2004332.63'2042—dc22 2003021212

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

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To Elizabeth—my friend, my companion, my wife

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Acknowledgments

Iwould like to thank Mike Burgess, Rod Hare, and Mitchell Duncan, who took time out of their busy schedules to read and critique the early manuscripts of this book. Their efforts transformed the original terse

descriptions of computer code and the often rambling musings and thought processes of an engineer into a readable document having a rational flow for you, the reader.

Tools are very important in our technological age. I would like to thank TradeStation Technologies for their platform, which made the develop-ment of trading systems possible. I would also like to thank eSignal for making their platform available for indicator development and Chris Kryza for converting my code to eSignal Formula Script. Additionally, I would like to thank Steve Ward, who made the resources of NeuroShell Trader available, thus enabling readers to extend the usefulness of my indicators by using neural networks and genetic algorithms.

I would also like to thank Mike Barna for showing me how to apply the coin toss methodology to trading strategy evaluation.

J. F. E.

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Introduction

CHAPTER 1

CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

CHAPTER 6

CHAPTER 7

CHAPTER 8

CHAPTER 9

CHAPTER 10

CHAPTER 11

CHAPTER 12

CHAPTER 13

CHAPTER 14

CHAPTER 15

CHAPTER 16

CHAPTER 17

Contents

xi

The Fisher Transform 1

Trends and Cycles 11

Trading the Trend 21

Trading the Cycle 33

The CG Oscillator 47

Relative Vigor Index 55

Oscillator Comparison 63

Stochasticization and Fisherization of Indicators 67

Measuring Cycles 107

Adaptive Cycle Indicators 123

The Sinewave Indicator 151

Adapting to the Trend 165

Super Smoothers 187

Time Warp—Without Space Travel 213

Evaluating Trading Systems 227

Leading Indicators 231

Simplifying Simple Moving Average Computations 241

Conclusion 245 For More Information 247 Notes 249 Index 251

ix

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Introduction

“This is a synopsis of my book,” Tom said abstractly.

A s Sir Arthur C. Clarke has noted, any significantly advanced technol-ogy is indistinguishable from magic. The advances made in com-puter technology in the past two decades have been dramatic and

can qualify as nearly magical. The computer on my desk today is far more powerful than that which was available to the entire national defense sys-tem just 30 years ago. Software for traders, however, has not kept pace. Most of the trading tools available today are neither different from nor more complex than the simple pencil-and-paper calculations that can be achieved through the use of mechanical adding machines. True, these cal-culations are now made with blinding speed and presented in colorful and eye-grabbing displays, but the power and usefulness of the underlying pro-cedures have not changed. If anything, the relative power of the calcula-tions has diminished because the increased speed of information exchange and increased market capitalization have caused fundamental shifts in the technical character of the market. These shifts include increased volatility and shorter periods for the market swings.

Cybernetic Analysis for Stocks and Futures promises to bring magic to the art of trading by introducing wholly new digital signal-processing tech-niques. The application of digital signal processing offers the advantage of viewing old problems from a new perspective. The new perspective gained by digital signal processing has led me to develop some profoundly effective new trading tools. The advances in trading tools, along with the continuing advancements in hardware capabilities, virtually ensure the continued ap-plication of digital signal processing in the future. Traders who master the new concepts, therefore, will find themselves at a great advantage when

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xii Introduction

approaching the volatile market of the twenty-first century. If you like code, you will love this book. Every new technique, indicator, and automatic trad-ing system is defined in exquisite detail in both EasyLanguage code for use in TradeStation and in eSignal Formula Script (EFS) code. They are also available as compiled DLLs to be run in NeuroShell trader.

Chapter 1 starts the wizardry off with a bang by challenging the con-ventional wisdom that market prices have a Gaussian probability density function (PDF). Just think about it. Do prices really have several events separated by a standard deviation from the mean across the screen as you would expect with a Gaussian PDF? Absolutely not! If the PDF is not Gaussian, then attaching significance to the one-sigma points in trading systems is, at best, just plain wrong. I show you how to establish an approx-imate Gaussian PDF through the application of the Fisher transform.

I derive a new zero-lag Instantaneous Trendline in Chapter 2. By divid-ing the market into a trend component and a cycle component, I create a zero-lag cycle oscillator from the derivation. These results are put to work by designing an automatic trend-following trading strategy in Chapter 3 and an automatic cycle-trading strategy in Chapter 4.

Several new oscillators are then derived. These include the CG Oscillator in Chapter 5 and the Relative Vigor Index (RVI) in Chapter 6. The performance of the Cyber Cycle Oscillator, the CG Oscillator, and the RVI are compared in Chapter 7. Noting that a favorite technical analysis tool is the Stochastic Relative Strength Index (RSI), where the RSI curve is sharp-ened by taking the Stochastic of it, I then show you in Chapter 8 how to enhance the oscillators by taking the Stochastic of them and also applying the Fisher transform.

In Chapter 9 I give an all-new exciting method of measuring market cycles. Using the Hilbert transform, a fast-reacting method of measuring cycles is derived. The validity and accuracy of these measurements are then demonstrated using several stressing theoretical waveforms. In Chapter 10 I then show you how to use the measured Dominant Cycle length to make standard indicators automatically adaptive to the measured Dominant Cycle. This adaptation makes good indicators stand out and sparkle as outstanding indicators. In Chapter 11, the cycle component of the Dominant Cycle is synthesized from the cycle measurement and displayed as the Sinewave Indicator. The advantages of the Sinewave Indicator are that it can anticipate cyclic turning points and that it is not subject to whipsaw trades when the market is in a trend. I continue the theme of adapting to the measured Dominant Cycle in Chapter 12 by show-ing you how to use the measurement to design an automatic trend-following trading strategy. The performance of the strategies I disclose is on par with or exceeds that of commercially available strategies.

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Introduction xiii

Chapter 13 provides you with several types of filters that give vastly superior smoothing with a minimum penalty in lag. Computer code is pro-vided for these filters, as well as tables of coefficient values. Another way to obtain superior smoothing is through the use of Laguerre polynomials. Laguerre polynomials enable smoothing to be done using a very short amount of data, as I explain in Chapter 14.

One of the problems with using backtests of automatic trading strate-gies is that they don’t necessarily predict future performance. I describe a technique in Chapter 15 that will enable you to use the theory of probabil-ity to visualize how your trading strategy could perform. It also illustrates what historical parameters are important to make this assessment. In Chapter 16 I show you how to generate leading indicators, along with the penalty in increased noise that you must accept when these indicators are used. I conclude in Chapter 17 by showing you how to simplify the coding of simple moving averages (SMAs).

Many of the digital signal-processing techniques described in this book have been known and used in the physical sciences for many years. For example, Maximum Entropy Spectral Analysis (MESA) algorithm was orig-inally developed by geophysicists in their exploration for oil. The small amount of data obtainable from seismic exploration demanded a solution using a short amount of data. I successfully adapted this approach and pop-ularized it for the measurement of market cycles. More recently, the use of digital signal processing has exploded in consumer electronics, making devices such as CDs and DVDs possible. Today, complete radio receivers are constructed without the use of analog components. As we expand DSP use by introducing it to the field of trading, we will see that digital signal processing is an exciting new field, perfect for technically oriented traders. It allows us to generalize and expand the use of many traditionally used indicators as well as achieve more precise computations.

I begin each chapter with a Tom Swifty. Perhaps this is a testament to my adolescent sense of humor, but the idea is to anchor the concept of the chapter in your mind. A Tom Swifty is a play on words that follows an unvarying pattern and relies for its humor on a punning relationship between the way an adverb describes the speaker and at the same time refers significantly to the import of the speaker’s statement, as in, “I like

fuzzy bunnies,” said Tom acutely. The combinations are endless. Since this book contains magic, perhaps I should have selected Harry Potter as a hero rather than Tom Swift.

Throughout this book my objective is to not only describe new tech-niques and tools but also to provide you the means to make your trading more profitable and therefore more pleasurable.

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Futures

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CHAPTER 1

The Fisher Transform

“I don’t see any chance of a market recovery,”

said Tom improbably.

The focus of my research for more than two decades has been directed toward applying my background in engineering and signal processing to the art of trading. The goal of this book is to share the

results of this research with you. Throughout the book I will demonstrate new methods for technical analysis of stocks and commodities and ways to code them for maximum efficiency and effectiveness. I will discuss meth-ods for modeling the market to help categorize market activity. In addition to new indicators and automatic trading systems, I will explain how to turn good-performing traditional indicators into outstanding adaptive indica-tors. The trading systems that subsequently evolve from this analysis will seriously challenge, and often exceed, the consistent performance and profit-making capabilities of most commercially available trading systems. While much of what is covered in this book breaks new ground, it is not simply innovation for innovation’s sake. Rather, it is intended to challenge conventional wisdom and illuminate the shortcomings of many prevailing approaches to systems development.

In this chapter we plunge right into an excellent example of challenging conventional wisdom. I know at least a dozen statistically based indicators that reference “the one-sigma point,” “the three-sigma point,” and so on. Sigma is the standard deviation from the mean. In order to have a standard deviation from the mean, one must know the probability density function (PDF). A Gaussian, or Normal, PDF is almost universally assumed. A Gaussian PDF is the familiar bell-shaped curve used to describe IQ distribu-tion in the population and a host of other statistical descriptions. The Gaussian PDF has long “tails” that describe events that have a wide deviation

1

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2 CYBERNETIC ANALYSIS FOR STOCKS AND FUTURES

from the mean with relatively low probability. With a Gaussian PDF, 68.26 percent of all occurrences fall within plus or minus one standard deviation from the mean, 95.44 percent of occurrences fall within plus or minus two standard deviations, and 99.73 percent of all occurrences fall within plus or minus three deviations. In other words, the majority of all cases fall within the one-sigma “boundary” with a Gaussian PDF. If an event falls outside the one-sigma level, then certain inferences have been drawn about what can happen in the future.

The real question here is whether the Gaussian PDF can be used to reli-ably describe market activity. You can easily answer that question yourself. Just think about the way prices look on a bar chart. Do you see only 68 per-cent of the prices clustered near the mean price? That is, do you see 32 per-cent of the prices separated by more than one deviation from the mean? And, do you see prices spike away from the mean nearly 5 percent of the time by two standard deviations? How often do you even see price spikes at all? If you don’t see these deviations, a Gaussian PDF is not a good assumption.

The Fisher transform is a simple mathematical process used to convert any data set to a modified data set whose PDF is approximately Gaussian. Once the Fisher transform is computed, we can then analyze the trans-formed data set in terms of its deviation from the mean.

The Commodity Channel Index (CCI), developed by Donald Lambert, is an example of reliance on the Gaussian PDF assumption. The equation to compute the CCI is

Price − Moving AverageCCI = �� (1.1)

0.015 * Deviation

Deviation is computed from the difference of prices and moving aver-age values over a period. The period of the moving average over which the computation is done is selectable by the user. The CCI can be viewed as the current deviation normalized to the standard deviation. But what gives with the 0.015 term? Well, conveniently enough, the reciprocal of 0.015 is 66.7, which is close enough to one standard deviation of a Gaussian PDF for most technical analysis work. The premise is that if prices exceed a standard deviation, they will revert to the mean. Therefore, the common rules are to sell if the CCI exceeds +100 and buy if the CCI is less than −100. Needless to say, the CCI can be improved substantially through the use of the Fisher transform.

Suppose prices behave as a square wave. If you tried to use the price crossing a moving average as a trading system, you would be destined for failure because the price has already switched to the opposite value by the time the movement is detected. There are only two price values. Therefore,

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3 The Fisher Transform

FIGURE 1.1 The Probability Distribution of a Square Wave Has Only Two Values

the probability distribution is 50 percent that the price will be at one value or the other. There are no other possibilities. The probability distribution of the square wave is shown in Figure 1.1. Clearly, this probability function does not remotely resemble a Gaussian probability distribution.

There is no great mystery about the meaning of a probability density or how it is computed. It is simply the likelihood the price will assume a given value. Think of it this way: Construct any waveform you choose by arrang-ing beads strung on a series of parallel horizontal wires. After the wave-form is created, turn the frame so the wires are vertical. All the beads will fall to the bottom, and the number of beads on each wire will stack up to demonstrate the probability of the value represented by each wire.

I used a slightly more sophisticated computer code, but nonetheless the same idea, to create the probability distribution of a sinewave in Figure 1.2. In this case, I used a total of 10,000 “beads.” This PDF may be surpris-ing, but if you stop and think about it, you will realize that most of the sam-pled data points of a sinewave occur near the maximum and minimum extremes. The PDF of a simple sinewave cycle is not at all similar to a Gaussian PDF. In fact, cycle PDFs are more closely related to those of a square wave. The high probability of a cycle being near the extreme values is one of the reasons why cycles are difficult to trade. About the only way to successfully trade a cycle is to take advantage of the short-term coherency and predict the cyclic turning point.

The Fisher transform changes the PDF of any waveform so that the transformed output has an approximately Gaussian PDF. The Fisher trans-form equation is

1 + x y = 0.5 * ln � (1.2)� 1 − x �

Where x is the input y is the output ln is the natural logarithm

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FIGURE 1.2 Sinewave Cycle PDF Does Not Resemble a Gaussian PDF

The transfer function of the Fisher transform is shown in Figure 1.3. The input values are constrained to be within the range −1 < X < 1.

When the input data is near the mean, the gain is approximately unity. For example, go to x = 0.5 in Figure 1.3. There, the Y value is only slightly larger than 0.5. By contrast, when the input approaches either limit within the

FIGURE 1.3 The Nonlinear Transfer of the Fisher Transform Converts Inputs (x Axis) to Outputs (y Axis) Having a Nearly Gaussian PDF

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FIGURE 1.4 The Fisher-Transformed Sinewave Has a Nearly Gaussian PDF Shape

range, the output is greatly amplified. This amplification accentuates the largest deviations from the mean, providing the “tail” of the Gaussian PDF. Figure 1.4 shows the PDF of the Fisher-transformed output as the familiar bell-shaped curve, compared to the input sinewave PDF. Both have the same probability at the mean value. The transformed output PDF is nearly Gaussian, a radical change from the sinewave PDF.

I measured the probability distribution of U.S. Treasury Bond futures over a 15-year span from 1988 to 2003. To make the measurement, I created a normalized channel 10 bars long. The normalized channel is basically the same as a 10-bar Stochastic Indicator. I then measured the price location within that channel in 100 bins and counted up the number of times the price was in each bin. The results of this probability distribution measure-ment are shown in Figure 1.5. This actual probability distribution more closely resembles the PDF of a sinewave rather than a Gaussian PDF. I then increased the length of the normalized channel to 30 bars to test the hypoth-esis that the sinewave-like probability distribution is only a short-term phe-nomenon. The resulting probability distribution is shown in Figure 1.6. The probability distributions of Figures 1.5 and 1.6 are very similar. I will leave it to you to extend the probability analysis to any market of your choice. I pre-dict you will get substantially similar results.

So what does this mean for trading? If the prices are normalized to fall within the range from −1 to +1 and subjected to the Fisher transform, extreme price movements are relatively rare events. This means the turn-ing points can be clearly and unambiguously identified. The EasyLanguage

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FIGURE 1.5 Probability Distribution of Treasury Bond Futures in a 10-Bar Channel over 15 Years

FIGURE 1.6 Probability Distribution of Treasury Bond Futures in a 30-Bar Channel over 15 Years

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Inputs: Price((H+L)/2), Len(10);

Vars: MaxH(0), MinL(0), Fish(0);

MaxH = Highest(Price, Len);MinL = Lowest(Price, Len);

Value1 = .5*2*((Price - MinL)/(MaxH - MinL) - .5) + .5*Value1[1];

If Value1 > .9999 then Value1 = .9999;If Value1 < -.9999 then Value1 = -.9999;

Fish = 0.25*Log((1 + Value1)/(1 - Value1)) + .5*Fish[1];

Plot1(Fish, “Fisher”);Plot2(Fish[1], “Trigger”);

FIGURE 1.7 EasyLanguage Code to Normalize Price to a 10-Day Channel and Compute Its Fisher Transform

code to do this is shown in Figure 1.7 and the eSignal Formula Script (EFS) code is shown in Figure 1.8. Value1 is a function used to normalize price within its last 10-day range. The period for the range is adjustable as an input. Value1 is centered on its midpoint and then doubled so that Value1 will swing between the −1 and +1 limits. Value1 is also smoothed with an exponential moving average whose alpha is 0.5. The smoothing may allow Value1 to exceed the 10-day price range, so limits are introduced to pre-clude the Fisher transform from blowing up by having an input value larger than unity. The Fisher transform is computed to be the variable “Fish”. Both Fish and Fish delayed by one bar are plotted to provide a crossover system that identifies the cyclic turning points.

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/*****************************************************Title: Fisher Transform*****************************************************/

function preMain() {setStudyTitle(“Fisher Transform”);setCursorLabelName(“Fisher”, 0);setCursorLabelName(“Trigger”, 1);setDefaultBarFgColor(Color.blue, 0);setDefaultBarFgColor(Color.red, 1);

setDefaultBarThickness(2, 0);setDefaultBarThickness(2, 1);

}

var Value1 = null;var Value1_1 = 0;var Fish = null;var Fish_1 = 0;var vPrice = null;var aPrice = null;

function main(nLength) {var nState = getBarState();

if (nLength == null) nLength = 10;if (aPrice == null) aPrice = new Array(nLength);

if (nState == BARSTATE_NEWBAR && vPrice != null) {aPrice.pop();aPrice.unshift(vPrice);if (Value1 != null) Value1_1 = Value1;if (Fish != null) Fish_1 = Fish;

}

vPrice = (high() + low()) / 2;aPrice[0] = vPrice;

if (aPrice[nLength-1] == null) return;

var MaxH = high();var MinL = low();var temp;

FIGURE 1.8 EFS Code to Normalize Price to a 10-Day Channel and Compute Its Fisher Transform

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for(i = 0; i < nLength; ++i) {MaxH = Math.max(MaxH, aPrice[i]);

MinL = Math.min(MinL, aPrice[i]);}

Value1 = .5 * 2 * ((vPrice - MinL) / (MaxH - MinL) - .5) + .5 * Value1_1;

if(Value1 > .9999) Value1 = .9999;if(Value1 < -.9999) Value1 = -.9999;

Fish = 0.25 * Math.log((1 + Value1) / (1 - Value1)) + .5 * Fish_1;

return new Array(Fish, Fish_1);}

FIGURE 1.8 (Continued)

The Fisher transform of the prices within an eight-day channel is plot-ted below the price bars in Figure 1.9. Note that the turning points are not only sharp and distinct, but they also occur in a timely fashion so that prof-itable trades can be entered. The Fisher transform is also compared to a similarly scaled moving average convergence-divergence (MACD) indica-tor in Figure 1.9. The MACD is representative of conventional indicators whose turning points are rounded and indistinct in comparison to the Fisher transform. As a result of the rounded turning points, the entry and exit signals are invariably late.

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FIGURE 1.9 The Fisher Transform of Normalized Prices Has Very Sharp Turning Points When Compared to Conventional Indicators such as the MACD

KEY POINTS TO REMEMBER

• Prices almost never have a Gaussian, or Normal, probability distribution. • Statistical measures based on Gaussian probability distributions, such

as standard deviations, are in error because the probability distribu-tion assumption underlying the calculation is in error.

• The Fisher transform converts almost any input probability distribu-tion to be nearly a Gaussian probability distribution.

• The Fisher transform, when applied to indicators, provides razor-sharp buy and sell signals.

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CHAPTER 2

Cycles Trends and

“That took the wind out of my sails,” said Tom disgustedly.

To a trader, Trend Modes and Cycle Modes are synonymous with selec-tion of a trading strategy. In an uptrend the obvious strategy is to buy and hold. Similarly, in a downtrend the strategy is to sell and hold.

Conversely, the best strategy in a Cycle Mode is to top-pick and bottom-fish. Traders usually use some variant of moving averages to trade the Trend Mode and some oscillator to trade the Cycle Mode. In either case, the lag induced by the calculations is one of the biggest problems for a trader.

To an analyst, Trend Modes and Cycle Modes are best described by their frequency content. Prices in Trend Modes vary slowly with respect to time. Therefore, Trend Modes disregard high-frequency components and use only the slowly varying low-frequency components. Moving averages are low-pass filters that allow only the low-frequency components to pass to their output, and that is why they are effective for Trend Mode trading. Oscillators are high-pass filters that almost completely disregard the low-frequency components.

I will use these concepts to create a complementary oscillator and moving average. Most important, both the oscillator and the moving aver-age have essentially no lag. The elimination of lag is crucial to the trading indicators and systems developed from them in later chapters. I consider the creation of these zero-lag tools one of the most important develop-ments described in this book. Searching for zero-lag tools has long been the focus of my research, and I have used descriptors such as Instantaneous

Trendline in previous publications. The techniques I show you in this chap-ter are entirely new, even if the names are similar.

11

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I will start with the well-known exponential moving average (EMA) to derive an optimum mathematical description of Trend Mode and Cycle Mode components. The equation for an EMA is

Output = α * Input + (1 − α) * Output[1] (2.1)

Where α is a number less than 1 and greater than 0

In words, this equation means we take a fraction of the current price and add to it the filtered output one bar ago multiplied by the quantity (1 − α). With these coefficients, if the input is unchanging (zero frequency), the out-put will eventually converge to the input value. That is, this filter has unity gain at zero frequency. We can describe this filter in terms of its transfer response, which is the output divided by its input. By using Z transform nota-tion, we let Z−1 denote one bar of lag as a multiplicative operator. Doing this, the transfer response of Equation 2.1 can be solved using algebra as

Output αH(z) = � = �� (2.2)

Input 1 − (1 − α) * Z−1

We can test Equation 2.2 by letting Z−1 equal +1 (zero frequency). When we do this, it is easy to see that the numerator is equal to the denominator, and so the gain is unity. The high-frequency attenuation of this filter can be tested at the highest possible frequency, the Nyquist frequency, by letting Z−1

equal −1. Using daily samples, the highest frequency we can analyze is 0.5 cycles per day (a two-bar cycle). This is the Nyquist frequency for daily data. The two-bar cycle attenuation is [α. /(2 − α)]. The general attenuation response of the EMA as a function of the frequency is shown in Figure 2.1. The period of a cycle component in Figure 2.1 can be calculated as the reci-procal of frequency. For example, a frequency of 0.1 cycles per day corre-sponds to a 10-bar period for that cycle component.

In principle, all we have to do to create a high-pass filter is subtract the transfer response of the low-pass filter from unity. The logic is that a trans-fer response of 1 represents all frequencies, and subtracting the low-pass response from it leaves the high-pass response as a residual. However, there is one problem with this approach: The high-frequency attenuation of the low-pass filter of Equation 2.2 is not infinite (i.e., the transfer response is 0) at the Nyquist frequency. A finite high-frequency response in the low-pass filter will lead to a gain error in the transfer response of the high-pass filter. The finite attenuation problem is eliminated by averaging two sequential input samples rather than using only a single input sample. In this case, the transfer response of the averaged-input low-pass filter is

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13 Trends and Cycles

FIGURE 2.1 EMA Frequency Response (α = 0.05)

α �� * (1 + Z−1)2

H(z) = �� (2.3)1 − (1 − α) * Z−1

Equation 2.3 guarantees that the transfer response of the low-pass fil-ter will be 0 when Z−1 = −1. The general frequency response of the averaged-input EMA is shown in Figure 2.2.

The lag of a simple moving average is approximately half the average length. For example, a 21-bar moving average has a lag of 10 bars. The alpha of an equivalent EMA is related to the length of a simple moving average as

2α = �� (2.4)Length + 1

Using Equation 2.4, an EMA using α = 0.05 is equivalent to a 39-bar sim-ple moving average. A 39-day simple moving average has a 19-day lag, approximately half its length. Examination of Figure 2.3 shows that the very low-frequency lag of an EMA whose α = 0.05 is indeed 19 days. Although the lag decreases as frequency is increased, it is of little consequence because

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FIGURE 2.2 Smoothed-Input EMA Frequency Response (α = 0.05)

FIGURE 2.3 Smoothed-Input Lag Response (α = 0.05)

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= ��

= ��


the filtered amplitude is so small at these frequencies. The real impact of lag of all moving averages is the value of the lag at very low frequencies.

With Equation 2.3 we now have the capacity to construct a high-pass filter. We will subtract Equation 2.3 from unity as

α �� * (1 + Z−1)2

HP(z) = 1 − �� (2.5)1 − (1 − α) * Z−1

1 − (1 − α) * Z−1 − � α

� * (1 + Z−1)2

1 − (1 − α) * Z−1

� α1 − �� * (1 − Z−1)

21 − (1 − α) * Z−1

Sharper attenuation can be obtained by using higher-order filters. However, I have learned that higher-order filters not only have greater lag, but they also have transient effects that impress false artifacts on their out-puts. This is somewhat like ringing a bell: The ringing is more a function of the bell itself rather than a filtered response of a driving force. A reasonable compromise is the use of a second-order Gaussian filter. A second-order Gaussian low-pass filter can be generated by taking an EMA and immedi-ately taking another identical EMA of the first EMA. This can be represented by squaring the transfer response. We can therefore obtain a second-order Gaussian high-pass filter response by squaring Equation 2.5 as

� α1 − ��

2

* (1 − 2 * Z−1 + Z−2)2

HP(z) = �� (2.6)1 − 2 * (1 − α) * Z−1 + (1 − α)2 * Z−2

Equation 2.6 is converted to an EasyLanguage statement as

HPF = (1 − α/2)2 * (Price − 2 * Price[1] + Price[2]) + 2 * (1 − α) * HPF[1] − (1 − α)2 * HPF[2]; (2.7)

The transfer responses of Equations 2.6 and 2.7 (they are the same) are plotted in Figure 2.4.

Figure 2.4 shows that only frequency periods longer than 40 bars (fre-quency = 0.025 cycles per day) are significantly attenuated. Thus we have created a high-pass filter with a relatively sharp cutoff response. Since the output of this filter contains essentially no trending components, it must be the cycle component of price.

c02.qxd 2/2/04 10:44 AM Page 16


FIGURE 2.4 Transfer Response of a Second-Order High-Pass Gaussian Filter (α = 0.05)

The complementary low-pass filter that produces the Instantaneous Trendline is found by subtracting the high-pass components of Equation 2.6 from unity. Skipping over the tedious algebra to put both elements of this subtraction over a common denominator, the equation for the low-pass Instantaneous Trendline is

IT(z) =

α�4

2

�α�2

2

�3�

4 α2

�

��1 − 2 * (1 − α) * Z−1 + (1 − α)2Z−2

� � � �Z−1 − Z−2α − α −+(2.8)

Equation 2.8 is converted to an EasyLanguage statement as

InstTrend = (α − (α/2)2) * Price + (α2/2) * Price[1] − (α − 3α2/4) * Price[2]) + 2 * (1 − α) * InstTrend[1] − (1 − α)2 * InstTrend[2]; (2.9)

Figure 2.5 shows the attenuation of the Instantaneous Trendline filter and how only the low-frequency components are passed. The attenuation characteristic of the Instantaneous Trendline in Figure 2.5 is almost identi-cal to that of the EMA shown in Figure 2.2.

The most important feature of the Instantaneous Trendline is that it

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FIGURE 2.5 Frequency Response of the Instantaneous Trendline Filter (α = 0.05)

has zero lag. That’s right—zero lag! The lag is 0 because Instantaneous Trendline was created by subtracting the transfer response of a high-pass filter from unity. Since the high-pass filter has a very small amplitude at low frequencies, the resulting low-frequency lag of the difference is just the lag of unity, which is 0. Figure 2.6 shows the lag profile of the Instantaneous Trendline as a function of frequency. While the lag does increase to 13 bars at an approximate frequency of 0.005 cycles per day (200-day period), a fre-quency that low is more important to investors than to traders.

The importance of the zero lag feature of the Instantaneous Trendline is demonstrated by comparing its response to an EMA having an equivalent alpha. Figure 2.7 gives this comparison in response to real market data. It is clear that the two averages have about the same degree of smoothing, but that the Instantaneous Trendline has zero lag. If it is more convenient, you can think of the Instantaneous Trendline as a centered moving average. The major advantage of the Instantaneous Trendline compared to the cen-tered moving average is that it can be used up to the right edge of the chart. That means that real indicators and trading systems can be built using it as a component. It is also clear that the lag of the Instantaneous Trendline is so small that a trader can begin to think about creating indicators and trad-ing systems as a function of the price crisscrossing it. In later chapters we will develop such indicators and trading systems.

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FIGURE 2.6 Lag of the Instantaneous Trendline Filter (α = 0.05)

FIGURE 2.7 Instantaneous Trendline Has Much Less Lag than an EMA (α = 0.05)

c02.qxd 2/2/04 10:44 AM Page 19



• The Instantaneous Trendline has zero lag. • The Instantaneous Trendline has about the same smoothing as an EMA

using the same alpha. • An EMA is a low-pass filter. • Higher-order Gaussian filters are the equivalent of applying the EMA

multiple times. • Using filters higher than second order is not advisable because of the

ringing transient responses of the higher-order filters. • A complementary cycle oscillator to the Instantaneous Trendline ex-

ists as a second-order high-pass filter. • The lag of the complementary cycle oscillator is 0.

c02.qxd 2/2/04 10:44 AM Page 20

c03.qxd 2/2/04 10:44 AM Page 21

CHAPTER 3

Trading the Trend

“The market is going up,” said Tom trendedly.

Having an Instantaneous Trendline with zero lag (Equations 2.8 and 2.9) is a good beginning to generate a responsive trend-following system. The system would be even more responsive if it contained

a trigger that preceded the Instantaneous Trendline rather than following it and offering a confirming signal. A leading trigger can be generated by adding a two-day momentum of the Instantaneous Trendline to the Instan-taneous Trendline itself.

The rationale for the leading trigger is that adding the two-day momen-tum to the current value in a trend is predicting where the Instantaneous Trendline will be two days from now. When plotting the trigger on the cur-rent bar, the trigger must lead the Instantaneous Trendline by two bars. On a more mathematical level, the lag of the trigger is shown in Figure 3.1. The figure shows that the low-frequency lead is two bars and the worst-case lag occurs at a frequency of 0.25 cycles per day (a four-bar cycle period). The lag is of no concern because the attenuation of the Instantaneous Trendline (shown in Figure 2.5) makes the amplitude of the components in the vicin-ity of 0.25 cycles per day almost irrelevant to the overall response.

There is a price to pay for achieving the lead response of the trigger. That price is that leading functions cause a higher-frequency gain in the fil-ter instead of attenuation, which has a smoothing effect. Therefore, high-frequency gain causes the resulting transfer response to look more ragged than the original function. This is the case for any momentum function. The gain response of the trigger has a maximum of 9.5 dB at a frequency of 0.25 cycles per day, as shown in Figure 3.2. In this case, the gain does not

21

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FIGURE 3.1 Lead and Lag of the Trigger as a Function of Frequency

FIGURE 3.2 Gain Response of the Trigger

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23 Trading the Trend

severely affect the smoothness of the trigger because the Instantaneous Trendline has an attenuation of 26 dB at 0.25 cycles per day, as shown in Figure 2.5. Therefore, using both terms to compute the net attenuation, the worst-case high-frequency smoothing attenuation is still about 16 dB. This means the trigger will have about the same degree of smoothness as the Instantaneous Trendline.

The Instantaneous Trendline and the Trigger of the trend-following sys-tem are shown as indicators in Figure 3.3; the EasyLanguage code to create these indicator lines is shown in Figure 3.4, and the eSignal Formula Script (EFS) code is shown in Figure 3.5. The process for creating a trend-following trading system from the indicators is simple. One unique aspect of the code is that the ITrend is forced to be a finite impulse response (FIR)-smoothed version of price for the first seven bars of the calculation. This initialization is included to cause the ITrend to converge more rapidly to its correct value from the beginning transient. The strategy enters a long position when the trigger crosses over the Instantaneous Trendline and enters a short position when the trigger crosses under the Instantaneous Trendline. However, an effective trading system is more than following a simple set of indicators.

First, experience has shown that greater profits result from using limit orders rather than market orders or stop orders. Market orders are self-explanatory. Stop orders mean the market must be going in the direction of the trade before the order is filled. For example, for long-position trades, the stop order must be placed above the current price. Thus, the price must

FIGURE 3.3 Crossing of the Trigger and Instantaneous Trendline are Trading Signals

c03.qxd 2/2/04 10:44 AM Page 24


Inputs: Price((H+L)/2), alpha(.07);

Vars: Smooth(0), ITrend(0), Trigger(0);

ITrend = (alpha - alpha*alpha/4)*Price + .5*alpha*alpha*Price[1] - (alpha - .75*alpha*alpha)*Price[2] + 2*(1 - alpha)*ITrend[1] - (1 - alpha)*(1 - alpha)*Itrend[2];

If currentbar < 7 then ITrend = (Price + 2*Price[1] + Price[2]) / 4;

Trigger = 2*Itrend - ITrend[2];

Plot1(Itrend, “ITrend”);Plot2(Trigger, “Trig”);

FIGURE 3.4 EasyLanguage Code for the ITrend Indicator

increase from its current level before you get stopped into the long-position trade. This means you necessarily give up some of the profits you would otherwise have gotten if you had entered on a market order at the instant of your signal. You can lose additional profits from stop orders due to slippage. Slippage is the difference between your stop value and the price at which your order actually got filled. In fast markets slippage can be substantial. If limit orders are placed for the long position, the limit price must be below the current price. That is, the market must move against your anticipated trade before you get a fill. This means that if the price drops sufficiently so that your limit order is filled, you have captured additional profits if the price subsequently reverses and goes in the direction of your signal. Furthermore, if there is any slippage in filling the limit order, the slippage will be negative because it is going in the direction opposite to your intended trade. When the price turns around and goes in the direction of your signals, you have therefore captured the slippage as profit. In the EasyLanguage trading strategy code of Figure 3.6, I have set the level of the limit order to be 35 percent of the current bar’s range added onto the clos-ing price of the current bar (in the case of a short signal) or subtracted from the closing price of the current bar (in the case of a long signal). The 35 per-cent is the input variable RngFrac, and is an optimizable parameter.

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/*****************************************************Title: Instantaneous Trendline*****************************************************/

function preMain() {setPriceStudy(true);setStudyTitle(“Instantaneous Trendline”);setCursorLabelName(“IT”, 0);setDefaultBarThickness(2, 0);

}

var a = 0.05;var IT = 0;var IT1 = 0;var IT2 = 0;var Price = 0;var Price1 = 0;var Price2 = 0;

function main() {if (getBarState() == BARSTATE_NEWBAR) {

IT2 = IT1;IT1 = IT;Price2 = Price1;Price1 = Price;

}

Price = close();

IT = (a-((a/2)*(a/2)))*Price + ((a*a)/2)*Price1 - (a-(3*(a*a))/4)*Price2 + 2*(1-a)*IT1 - ((1-a)*(1-a))*IT2;

return (IT);}

FIGURE 3.5 EFS Code for the ITrend Indicator

Unfortunately, not all trading signals are perfect. In fact, with the crossover strategy that I have developed it is possible to be on the wrong side of the trade for a substantial period from time to time. For this reason, I have added a rule that if the price goes against your position by more than some percentage, the strategy will correct itself and automatically reverse to the opposite position. The percentage is supplied as the input variable

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Inputs: Price((H+L)/2), alpha(.07), RngFrac(.35), RevPct(1.015);

Vars: Smooth(0), ITrend(0), Trigger(0);

ITrend = (alpha - alpha*alpha/4)*Price + .5*alpha*alpha*Price[1] – (alpha - .75*alpha*alpha)*Price[2] + 2*(1 – alpha)*ITrend[1] – (1 - alpha)*(1 - alpha)*ITrend[2];

If currentbar < 7 then ITrend = (Price + 2*Price[1] + Price[2]) / 4;

Trigger = 2*Itrend - ITrend[2];

If Trigger Crosses Over ITrend then Buy Next Bar atClose – RngFrac*(High - Low) Limit;

If Trigger Crosses Under ITrend then Sell Short NextBar at Close + RngFrac*(High - Low) Limit;

If MarketPosition = 1 and Close < EntryPrice/RevPct then Sell Short Next Bar On Open;

If MarketPosition = -1 and Close > RevPct*EntryPrice then Buy Next Bar on Open;

FIGURE 3.6 EasyLanguage Code for the Instantaneous Trendline Trading Strategy

RevPct. RevPct is an optimizable parameter, but I find that the default value of 1.5 percent (RevPct = 1.015) is a relatively robust number. The same strategy for EFS code is given in Figure 3.7.

I applied the strategy code of Figures 3.6 and 3.7 to several currency futures because it is well known that currencies tend to trend. I addition-ally introduced a $2,500 money management stop to further avoid giving back accumulated profits. Doing this, I achieved the trading results shown in Table 3.1. The time span is on the order of a quarter century, and a rela-tively large number of trades are taken. The Instantaneous Trend Strategy consists of only a few independent parameters. Since the ratio of the num-ber of trades to the number of parameters is large and since the trading took place over a large time span, it is highly unlikely that the strategy has

c03.qxd 2/2/04 10:44 AM Page 27


/*****************************************************Title: ITrend Trading Strategy Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/27/2003Version: 1.0.0

======================================================Fix History:

06/27/2003 - Initial Release1.0.0

======================================================*****************************************************/

//External Variables

var grID = 0; var nBarCount = 0; var xOver = 0; var nStatus = 0; var nEntryPrice = 0; var nDirection = 0; var nLimitPrice = 0; var nAdj1 = null;

var aPriceArray = new Array(); var aITrendArray = new Array();

//== PreMain function required by eSignal to set_things up

function preMain() {var x;

setPriceStudy(true);setStudyTitle(“ITrend Strategy”);setCursorLabelName(“ITrend”, 0);setCursorLabelName(“Trig”, 1);

(continued)

FIGURE 3.7 EFS Code for the Instantaneous Trendline Trading Strategy

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setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );

//initialize arraysfor (x=0; x<10; x++) {

aPriceArray[x] = 0.0; aITrendArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha, RngFrac, RevPct ) {var x;var nPrice;

if (getCurrentBarIndex() == 0) return;

//initialize parameters if necessaryif ( Alpha == null ) {

Alpha = 0.07;}if ( RngFrac == null ) {

RngFrac = 0.35;}if ( RevPct == null ) {

RevPct = 1.015;}

// study is initializingif (getBarState() == BARSTATE_ALLBARS) {return null;

}

if (nAdj1 == null) nAdj1 = (high()-low()) * 0.20;

//on each new bar, save array valuesif ( getBarState() == BARSTATE_NEWBAR ) {

nBarCount++;

aPriceArray.pop();aPriceArray.unshift( 0 );


c03.qxd 2/2/04 10:44 AM Page 29


aITrendArray.pop();aITrendArray.unshift( 0 );

}

nPrice = ( high()+low() ) / 2;aPriceArray[0] = nPrice;

if (aPriceArray[2] == 0) return;

if ( nBarCount < 7 ) {aITrendArray[0] = (nPrice

+ 2*aPriceArray[1] + aPriceArray[2])/4;

}else {

aITrendArray[0] = (Alpha - Alpha*Alpha/4)*nPrice + 0.5*Alpha*Alpha*aPriceArray[1] - (Alpha - 0.75*Alpha*Alpha) * aPriceArray[2] + 2*(1-Alpha)*aITrendArray[1] - (1-Alpha)*(1-Alpha)*aITrendArray[2];

}

if (aITrendArray[2] == 0) return;

nTrig = 2 * aITrendArray[0] - aITrendArray[2];

nStatus = 0;if ( Strategy.isLong() ) nStatus = 1;if ( Strategy.isShort() ) nStatus = -1;

var bReverseTrade = false;if ( nStatus == 1 && close()

< (nEntryPrice/RevPct) ) {ReverseToShort();bReverseTrade = true;

} else if ( nStatus == -1 && close() > (RevPct*nEntryPrice) ) {

ReverseToLong();bReverseTrade = true;

(continued)


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}//check for new signals

if (bReverseTrade == false) {if ( nTrig > aITrendArray[0] ) {

if ( xOver == -1 && nStatus != 1) {nLimitPrice = Math.max(low(), (close()

- ( high()-low() )*RngFrac));LongLimit( nLimitPrice );nDirection = 1;

}xOver = 1;

} else if ( nTrig < aITrendArray[0] ) {if ( xOver == 1 && nStatus != -1) {

nLimitPrice = Math.min(high(), (close() + ( high()-low() )*RngFrac));

ShortLimit( nLimitPrice );nDirection = -1;

}xOver = -1;

}}

if (!isNaN( aITrendArray[0] ) ) {return new Array( aITrendArray[0],_

nTrig );}

}

function LongLimit( nPrice ) {Strategy.doLong(“Long”, Strategy.LIMIT,_

Strategy.THISBAR, Strategy.DEFAULT,_nPrice );

nEntryPrice = nPrice;drawShapeRelative(0, low()-nAdj1, Shape.UPARROW,_

““, Color.lime, Shape.ONTOP, gID());return;

}

function ShortLimit( nPrice ) {Strategy.doShort(“Short”, Strategy.LIMIT,_

Strategy.THISBAR, Strategy.DEFAULT,_nPrice );


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nEntryPrice = nPrice;debugPrintln(getCurrentBarIndex()

+ “ short “ + nPrice);drawShapeRelative(0, high()+nAdj1,Shape.DOWNARROW,_

““, Color.maroon, Shape.ONTOP, gID());return;

}

function ReverseToLong() {Strategy.doLong(“Reverse to Long”,_

Strategy.MARKET, Strategy.NEXTBAR,_Strategy.DEFAULT );

DrawShapeRelative(1, low(1)-nAdj1,_Shape.UPARROW, ““, Color.lime,_Shape.ONTOP, gID());

nEntryPrice = open(1);nStatus = 1;nDirection = 0;nLimitPrice = 0;return;

}

function ReverseToShort() {Strategy.doShort(“Reverse to Short”,_

Strategy.MARKET, Strategy.NEXTBAR,_Strategy.DEFAULT );

drawShapeRelative(1, high(1)+nAdj1,_Shape.DOWNARROW, ““, Color.maroon,_Shape.ONTOP, gID());

nEntryPrice = open(1);nStatus = -1;nDirection = 0;nLimitPrice = 0;return;

}

//== gID function assigns unique identifier to_graphic/text routines

function gID() {grID ++;return( grID );

}


c03.qxd 2/2/04 10:44 AM Page 32


TABLE 3.1 Sample Trading Results Using the Instantaneous Trendline Strategy

Number Percent Profit Max Future Net Profit of Trades Profitable Factor DD

EC (4/81–3/03) $201,812 230 42.2% 1.89 ($26,775) JY (9/81–3/03) $221,312 229 48.5% 2.50 ($11,712) SF (6/76–3/03) $129,175 337 45.1% 1.52 ($15,387)

been curve fitted. Curve fitting is a weakness of many technical analysis trading strategies.

Please allow me to brag about the Instantaneous Trendline Strategy. (Perhaps it is not bragging, because as Muhammed Ali said, “It ain’t brag-ging if you can really do it.”) The performance results of this strategy are comparable to, or exceed, the performance of commercial systems costing thousands of dollars. You can create synthetic equity growth curves using the established percentage of profitable trades and profit factors. This is explained in Chapter 15. You will find the equity growth trading the cur-rencies in Table 3.1 to be remarkably consistent.


• The Instantaneous Trendline has zero lag. • The Instantaneous Trendline has about the same smoothing as an

exponential moving average (EMA) using the same alpha. • The smoothing enables the use of a trading trigger that has a two-bar

lead. • Trading signals are generated by the crossing of the Trigger line and the

Instantaneous Trendline. • Trade entries are made on limit orders to capture a larger range of the

trade and to eliminate slippage losses. • Major losses are avoided by recognizing when a trade is on the wrong

side and reversing position. • The Instantaneous Trendline Strategy can be optimized for application

to many stocks and commodity markets.

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CHAPTER 4

the Cycle Trading

“It happens again and again,” said Tom periodically.

Equation 2.5 described a high-pass filter that isolated the cycle mode components. Essentially all that need be done to generate a cycle-based indicator is to plot the results of this equation. However, some

smoothing is required to remove the two-bar and three-bar components that detract from the interpretation of the cyclic signals. These compo-nents can be removed with a simple finite impulse response (FIR)1 low-pass filter as

Smooth = (Price + 2 * Price[1] + 2 * Price[2] + Price[3])/6; (4.1)

The lag of the Smooth filter of Equation 4.1 is 1.5 bars at all frequen-cies. Figure 4.1 demonstrates that the Smooth filter eliminates the two- and three-bar cycle components. The Smooth filter is to be used as an addi-tional filter to remove the distracting very-high-frequency components, thus creating an indicator that is easier to interpret for trading.

The EasyLanguage code to make a cycle component indicator is given in Figure 4.2 and the eSignal Formula Script (EFS) code is given in Figure 4.3. I call this the Cyber Cycle Indicator. After the inputs and variables are defined, the smoothing filter of Equation 4.1 and the high-pass filter of Equation 2.7 are computed. They are followed by an initialization condition that facilitates a rapid convergence at the beginning of the input data. A trading trigger signal is created by delaying the cycle by one bar.

Trading the Cyber Cycle Indicator is straightforward. Buy when the Cycle line crosses over the Trigger line. You are at the bottom of the cycle

33

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FIGURE 4.1 A Four-Element FIR Filter Eliminates Two- and Three-Bar Cycles

Inputs: Price((H+L)/2), alpha(.07);

Vars: Smooth(0), Cycle(0);

Smooth = (Price + 2*Price[1] + 2*Price[2] + Price[3])/6;

Cycle = (1 - .5*alpha)*(1 - .5*alpha)*(Smooth - 2*Smooth[1] + Smooth[2]) + 2*(1 - alpha)*Cycle[1] - (1 - alpha)*(1 - alpha)*Cycle[2];

If currentbar < 7 then Cycle = (Price - 2*Price[1] + Price[2]) / 4;

Plot1(Cycle, “Cycle”);Plot2(Cycle[1], “Trigger”);

FIGURE 4.2 EasyLanguage Code for the Cyber Cycle Indicator

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35 Trading the Cycle

/*****************************************************Title: Cyber Cycle*****************************************************/

function preMain() {setStudyTitle(“High Pass Filter”);setCursorLabelName(“HPF”,0);setDefaultBarThickness(2, 0);

}

var a = 0.07;var HPF = 0;var HPF1 = 0;var HPF2 = 0;var Price = 0;var Price1 = 0;var Price2 = 0;

function main() {if (getBarState() == BARSTATE_NEWBAR) {

HPF2 = HPF1;HPF1 = HPF;Price2 = Price1;Price1 = Price;

}

Price = close();

HPF = ((1-(a/2))*(1-(a/2))) * (Price - 2*Price1 + Price2) + 2*(1-a)*HPF1 - ((1-a)*(1-a))*HPF2;

return (HPF);}

FIGURE 4.3 EFS Code for the Cyber Cycle Indicator

at this point. Sell when the Cycle line crosses under the Trigger line. You are at the top of the cycle in this case. Figure 4.4 illustrates that each of the major turning points is captured by the Cycle line crossing the Trigger line. To be sure, there are crossings at other than the cyclic turning points. Many of these can be eliminated by discretionary traders using their experience or others of their favorite tools.

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FIGURE 4.4 The Cyber Cycle Indicator Catches Every Significant Turning Point

One of the more interesting aspects of the Cyber Cycle is that it was developed simultaneously with the Instantaneous Trendline. They are opposite sides of the same coin because the total frequency content of the market being analyzed is in one indicator or the other. This is important because the conventional methods of using moving averages and oscilla-tors can be dispensed with. The significance of this duality is demonstrated in Figure 4.5.

A low-lag four-bar weighted moving average (WMA) is plotted in Figure 4.5 for comparison with the action of the Instantaneous Trendline. Note that each time the WMA crosses the Instantaneous Trendline the Cyber Cycle Oscillator is also crossing its zero line. Since there is essentially no lag in the Instantaneous Trendline we can, for the first time, use an indicator overlay on prices in exactly the same way we have traditionally used oscillators. That is, when the prices cross the Instantaneous Trendline you can start to prepare for a reversal when prices reach a maximum excursion from the Instantaneous Trendline. Since there is only a small lag in the Instantaneous Trendline, it represents a short-term mean of prices. This being the case, we can use the old principle that prices revert to their mean.

But what is the best way to exploit the mean reversion? The false sig-nals arising from use of the Cyber Cycle are more problematic for automatic trading systems. The first thing that must be understood about indicators is that they are invariably late. No indicator can precede an event from which it is derived. This is particularly important when trading short-term cycles.

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FIGURE 4.5 The Instantaneous Trendline and Cyber Cycle Oscillator are Duals

We need an indicator that predicts the turning point so the trade can be made at the turning point or even before it occurs. In the code of Figure 4.2 we know we induce 1.5 bars of lag due to the calculation of Smooth. The cycle equation contributes some small amount of lag also, perhaps half a bar. The Trigger lags the Cycle by one bar, so that their crossing introduces at least another bar of lag. Finally, we can’t execute the trade until the bar after the signal is observed. In total, that means our trade execution will be at least four bars late. If we are working with an eight-bar cycle, that means the signal will be exactly wrong. We could do better to buy when the signal says sell, and vice versa.

The difficulties arising from the lag suggest a way to build an automatic trading strategy. Suppose we choose to use the trading signal in the oppo-site direction of the signal. That will work if we can introduce lag so the correct signal will be given in the more general case, not just the case of an eight-bar cycle. Figure 4.6 is the EasyLanguage code for the Cyber Cycle strategy. It starts exactly the same as the Cyber Cycle Indicator. I then introduce the variable Signal, which is an exponential moving average of the Cycle variable. The exponential moving average generates the desired lag in the trading signal. As derived in Rocket Science for Traders,2 the rela-tionship between the alpha of an exponential moving average and lag is

1α = � (4.2)Lag + 1

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Inputs: Price((H+L)/2), alpha(.07), Lag(9);

Vars: Smooth(0), Cycle(0), alpha2(0), Signal(0);




alpha2 = 1 / (Lag + 1);Signal = alpha2*Cycle + (1 - alpha2)*Signal[1];

If Signal Crosses Under Signal[1] then Buy Next_Bar on Open;

If Signal Crosses Over Signal[1] then Sell Short Next_Bar on Open;

If MarketPosition = 1 and PositionProfit < 0 and BarsSinceEntry > 8 then Sell This Bar;

If MarketPosition = -1 and PositionProfit < 0 and BarsSinceEntry > 8 then Buy To Cover This Bar;

FIGURE 4.6 EasyLanguage Code for the Cyber Cycle Trading Strategy

This relationship is used to create the variable alpha2 in the code and the variable Signal using the exponential moving average.

The trading signals using the variable Signal crossing itself delayed by one bar are exactly the opposite of the trading signals I would have used if there were no delay. But, since the variable Signal is delayed such that the net delay is less than half a cycle, the trading signals are correct to catch the next cyclic reversal.

The idea of betting against the correct direction by waiting for the next cycle reversal can be pretty scary because that reversal may “never” happen because the market takes off in a trend. For this reason I included two lines

c04.qxd 2/2/04 10:45 AM Page 39


of code that are escape mechanisms if we were wrong in our entry signal. These last two lines of code in Figure 4.6 reverse the trading position if we have been in the trade for more than eight bars and the trade has an open position loss.

The EFS code for the Cyber Cycle Trading Strategy is given in Figure 4.7. The trading strategy of Figures 4.6 and 4.7 was applied to Treasury

Bond futures because this contract tends to cycle and not stay in a trend for long periods. The performance response from January 4, 1988 to March 3, 2003, a period in excess of 15 years, produced the results shown in Table 4.1. These performance results, and the consistent equity growth depictedin Figure 4.8, exceed the results of most commercially available trading systems designed for Treasury Bonds.

/*****************************************************Title: Cyber Cycle Trading Strategy Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/27/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var grID = 0;var nBarCount = 0;var nStatus = 0; //0=flat, -1=short,_

1=long//var nTrigger = 0; //buy/sell on next openvar nBarsInTrade = 0;var nEntryPrice = 0;

(continued)

FIGURE 4.7 EFS Code for the Cyber Cycle Trading Strategy

c04.qxd 2/2/04 10:45 AM Page 40


var nAdj1 = 0;var nAdj2 = 0;

var aPriceArrayvar aSmoothArrayvar aCycleArrayvar aSignalArray

= new Array();= new Array();= new Array();= new Array();



//setPriceStudy( true );setStudyTitle(“CyberCycle Strategy”);

//setShowCursorLabel( false );

setCursorLabelName(“Signal “, 0);setCursorLabelName(“Signal1”, 1);

setDefaultBarFgColor(Color.blue, 0);setDefaultBarFgColor(Color.red, 1);


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aSignalArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha, Lag ) {var x;var nPrice;var nAlpha2;


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if (getCurrentBarIndex() == 0) return;


Alpha = 0.07;}

if ( Lag == null ) {Lag = 20;

}


}


nBarCount++;nBarsInTrade++;

//variables for image alignmentnAdj1 = (high()-low()) * 0.20;nAdj2 = (high()-low()) * 0.35;


aSmoothArray.pop();aSmoothArray.unshift( 0 );

aCycleArray.pop();aCycleArray.unshift( 0 );

aSignalArray.pop();aSignalArray.unshift( 0 );

}

//Cyber Cycle formulanPrice = ( high()+low() ) / 2;

(continued)


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aPriceArray[0] = nPrice;

if (aPriceArray[3] == 0) return;

aSmoothArray[0] = ( aPriceArray[0] + 2*aPriceArray[1] + 2*aPriceArray[2] + aPriceArray[3] ) / 6;

if ( nBarCount < 7 ) {aCycleArray[0] = ( aPriceArray[0]

- 2*aPriceArray[1] + aPriceArray[2] ) / 4;

}else {

aCycleArray[0] = ( 1 - 0.5*Alpha ) * ( 1 - 0.5*Alpha ) * ( aSmoothArray[0] - 2*aSmoothArray[1] + aSmoothArray[2] ) + 2*( 1-Alpha ) * aCycleArray[1] - ( 1-Alpha ) * ( 1-Alpha ) * aCycleArray[2];

}

//create the actual trading signalsnAlpha2 = 1 / (Lag + 1 );aSignalArray[0] = nAlpha2 * aCycleArray[0]

+ ( 1.0 - nAlpha2 ) * aSignalArray[1];

//process our trading strategy code//=================================

nStatus = 0;if (Strategy.isLong() == true) nStatus = 1;if (Strategy.isShort() == true) nStatus = -1;

//currently not in a trade so look for a triggerif ( nBarCount > 10 && nStatus == 0 ) {

//signal cross down - we buyif ( aSignalArray[0] < aSignalArray[1]_


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&& aSignalArray[1] >= aSignalArray[2] ) {

goLong();}//signal cross up - we sellif ( aSignalArray[0] > aSignalArray[1]_

&& aSignalArray[1] <= aSignalArray[2] ) {

goShort();}

}//currently in a trade so look for profit stop_

or reversalelse if ( nBarCount > 10 && nStatus != 0 ) {

if ( nStatus == 1 ) { //in a long trade//if trade is unprofitable after_

8 bars, exit positionif ( close() - nEntryPrice

< 0 && nBarsInTrade > 8 ) {closeLong();

}//otherwise, check for trigger in_

other directionif ( aSignalArray[0]

> aSignalArray[1]_&& aSignalArray[1] <= aSignalArray[2] ) {

goShort();}

} else if ( nStatus == -1 ) { //in a_short trade

//if trade is unprofitable after_8 bars, exit position

if ( nEntryPrice - close() < 0_&& nBarsInTrade > 8 ) {

closeShort();}//otherwise, check for trigger in_

other directionif ( aSignalArray[0]

< aSignalArray[1]_(continued)


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&& aSignalArray[1] >= aSignalArray[2] ) {

goLong();}

}}

return new Array(aSignalArray[0],_aSignalArray[1]);

}

//enter a short tradefunction goShort() {

drawShapeRelative(1, aSignalArray[1],_Shape.DOWNARROW, ““, Color.maroon, Shape.ONTOP|Shape.BOTTOM,gID());Strategy.doShort(“Short Signal”,_Strategy.MARKET, Strategy.NEXTBAR,Strategy.DEFAULT );

nStatus = -1; nEntryPrice = open(1); nBarsInTrade = 1;

}

//exit a short tradefunction closeShort() {

drawShapeRelative(-0, aSignalArray[0],_Shape.DIAMOND, ““, Color.maroon, Shape.ONTOP|Shape.TOP, gID());

Strategy.doCover(“Cover Short”,_Strategy.MARKET, Strategy.THISBAR,_Strategy.ALL );

nStatus = 0;nEntryPrice = 0;

}

//enter a long tradefunction goLong() {

drawShapeRelative(1, aSignalArray[1],_


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Shape.UPARROW, ““, Color.lime, Shape.ONTOP|Shape.TOP, gID());

Strategy.doLong(“Long Signal”, Strategy.MARKET,_Strategy.NEXTBAR, Strategy.DEFAULT );

nStatus = 1; nEntryPrice = open(1); nBarsInTrade = 1;

}

//exit a long tradefunction closeLong() {

drawShapeRelative(0, aSignalArray[0],_Shape.DIAMOND, ““,Color.lime, Shape.ONTOP|Shape.BOTTOM, gID());Strategy.doSell(“Sell Long”, Strategy.MARKET,_

Strategy.THISBAR, Strategy.ALL );nStatus = 0;nEntryPrice = 0;

}

//== gID function assigns unique identifier tographic/text routines

function gID() {grID ++;return( grID );

}


TABLE 4.1 Fifteen-Year Performance of the Cyber Cycle Trading System Trading Treasury Bond Futures

Net profit $93,156 Number of trades 430 Percent profitable 56.7% Profit factor 1.44 Max drawdown ($12,500) Profit/trade $216.64

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FIGURE 4.8 Cyber Cycle Trading System 15-Year Equity Growth Trading Treasury Bonds


• All indicators have lag. • The Instantaneous Trendline and the Cyber Cycle Indicator are com-

plementary. This enables traders to use indicators overlaid on prices the same way conventional oscillators are used.

• A viable cycle-based trading system delays the signal slightly less than a half cycle to generate leading turning point entry and exit signals.

• Major losses are avoided by recognizing when a trade is on the wrong side and reversing position.

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CHAPTER 5

The CG Oscillator

“Add up this list of n numbers and then divide the sum by n,”

said Tom meanly.

In this chapter I describe a new oscillator that is unique because it is smoothed and has essentially zero lag. The smoothing enables clear identification of turning points and the zero-lag aspect enables action

to be taken early in the move. This oscillator, which is the serendipitous result of my research into adaptive filters, has substantial advantages over conventional oscillators used in technical analysis. The CG in the name of the oscillator stands for the center of gravity of the prices over the window of observation.

The center of gravity (CG) of a physical object is its balance point. For example, if you balance a 12-inch ruler on your finger, the CG will be at its 6-inch point. If you change the weight distribution of the ruler by putting a paper clip on one end, then the balance point (i.e., the CG) shifts toward the paper clip. Moving from the physical world to the trading world, we can substitute the prices over our window of observation for the units of weight along the ruler. Using this analogy, we see that the CG of the win-dow moves to the right when prices increase sharply. Correspondingly, the CG of the window moves to the left when prices decrease.

The idea of computing the center of gravity arose from observing how the lags of various finite impulse response (FIR) filters vary according to the relative amplitude of the filter coefficients. A simple moving average (SMA) is an FIR filter where all the filter coefficients have the same value (usually unity). As a result, the CG of the SMA is exactly in the center of the filter. A weighted moving average (WMA) is an FIR filter where the most recent price is weighted by the length of the filter, the next most recent price is weighted by the length of the filter less 1, and so on. The weighting

47

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terms are the filter coefficients. The filter coefficients of a WMA describe the outline of a triangle. It is well known that the CG of a triangle is located at one-third the length of the base of the triangle. In other words, the CG of the WMA has shifted to the right relative to the CG of an SMA of equal length, resulting in less lag. In all FIR filters, the sum of the product of the coefficients and prices must be divided by the sum of the coefficients so that the scale of the original prices is retained.

The most general FIR filter is the Ehlers Filter,1 which can be written as

N

� ci * Pricei

N

� i = 0Ehlers Filter = �� (5.1)

ci

i = 0

The coefficients of the Ehlers Filter can be almost any measure of vari-ability. I have looked at momentum, signal-to-noise ratio, volatility, and even Stochastics and Relative Strength Index (RSI) values as filter coeffi-cients. One of the most adaptive sets of coefficients arose from video edge detection filters, and was the sum of the square of the differences between each price and each previous price. In any event, the result of using differ-ent filter coefficients is to make the filter adaptive by moving the CG of the coefficients.

While I was debugging the code of an adaptive FIR filter, I noticed that the CG itself moved in exact opposition to the price swings. The CG moves to the right when prices go up and to the left when prices go down. Measured as the distance from the most recent price, the CG decreased when prices rose and increased when they fell. All I had to do was to invert the sign of the CG to get a smoothed oscillator that was in phase with the price swings and had essentially zero lag.

The CG is computed in much the same way as we computed the Ehlers Filter. The position of the balance point is the summation of the product of position within the observation window times the price at that position divided by the summation of prices across the window. The mathematical expression for this calculation is

N

� (xi + 1) * Pricei

N

� i = 0CG = �� (5.2)

Pricei

i = 0

In this expression I added 1 to the position count because the count started with the most recent price at zero, and multiplying the most recent price by the position count would remove it from the computation. The

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49 The CG Oscillator

Inputs: Price((H+L)/2), Length(10);

Vars: count(0), Num(0), Denom(0), CG(0);

Num = 0; Denom = 0; For count = 0 to Length - 1 begin

Num = Num + (1 + count)*(Price[count]);Denom = Denom + (Price[count]);

End;If Denom <> 0 then CG = -Num/Denom + (Length + 1) / 2;

Plot1(CG, “CG”);Plot2(CG[1], “CG1”);

FIGURE 5.1 EasyLanguage Code to Compute the CG Oscillator

EasyLanguage code to compute the CG Oscillator is given in Figure 5.1 and the eSignal Formula Script (EFS) code is given in Figure 5.2.

In EasyLanguage, the notation Price[N] means the price N bars ago. Thus Price[0] is the price for the current bar. Counting for the location is backward from the current bar. In the code the summation is accomplished by recursion, where the count is varied from the current bar to the length of the observation window. The numerator is the sum of the product of the bar position and the price, and the denominator is the sum of the prices. Then the CG is just the negative ratio of the numerator to the denominator. A zero counter value for CG is established by adding half the length of the observation window plus 1. Since the CG is smoothed, an effective crossover signal is produced simply by delaying the CG by one bar.

An example of the CG Oscillator is shown in Figure 5.3. In this case, I selected the length to be an eight-bar observation window. It is clear that every major price turning point is identified with zero lag by the CG Oscillator and the crossovers formed by its trigger. Since the CG Oscillator is filtered and smoothed, whipsaws of the crossovers are minimized. The relative amplitudes of the cyclic swings are retained. The resemblance of the CG Oscillator to the Cyber Cycle Indicator of Chapter 4 is striking. I will compare all the oscillator type indicators in a later chapter.

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/*****************************************************Title: CG Oscillator Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/27/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/

//External Variables var nPrice = 0; var nCG = 0;

var aPriceArray = new Array(); var aCGArray = new Array();



setPriceStudy(false);setStudyTitle(“CG Osc”);setCursorLabelName(“CG”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );


aPriceArray[x] = 0.0;aCGArray[x] = 0.0;

FIGURE 5.2 EFS Code to Compute the CG Oscillator

c05.qxd 2/2/04 10:45 AM Page 51


}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;var nValue1;

//initialize parameters if necessaryif ( OscLength == null ) {

OscLength = 10;}


}



aCGArray.pop();aCGArray.unshift( 0 );

}


nNum = 0;nDenom = 0;

for ( x=0; x<OscLength; x++ ){nNum += ( 1.0 + x ) * ( aPriceArray[x] );

(continued)


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nDenom += ( aPriceArray[x] );}

if ( nDenom != 0 ) nCG = -nNum/nDenom + ( OscLength+1 )/2;

aCGArray[0] = nCG;

//return the calculated valuesif ( !isNaN( aCGArray[0] ) ) {

return new Array( aCGArray[0],_aCGArray[1] );

}

}


FIGURE 5.3 The CG Oscillator Accurately Identifies Each Price Turning Point

c05.qxd 2/2/04 10:45 AM Page 53


The appearance of the CG Oscillator varies with the selection of the observation window length. Ideally, the selected length should be half the dominant cycle length because half the dominant cycle fully captures the entire cyclic move in one direction. If the length is too long, the CG Oscillator is desensitized. For example, if the window length is one full dominant cycle, half the data pulls the CG to the right and the other half of the data pulls the CG to the left. As a result, the CG stays in the middle of the window and no motion of the CG Oscillator is observed. On the other hand, if the window length is too short, you are missing the benefits of smoothing. As a result of this case, the CG Oscillator contains higher-frequency components and is a little too nervous for profitable trading.


• The CG in an FIR filter is the position of the average price within the fil-ter window length.

• The CG moves toward the most recent bar (decreases) when prices rise and moves away from the most recent bar (increases) when prices fall. Thus the CG moves exactly opposite to the price direction.

• The CG Oscillator has essentially zero lag. • The CG Oscillator retains the relative cycle amplitude, similar to the

Cyber Cycle Indicator.

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c06.qxd 2/2/04 10:45 AM Page 55

CHAPTER 6

Relative Vigor Index

“Get to the back of the boat,” said Tom sternly.

This chapter describing the Relative Vigor Index (RVI) uses concepts dating back over three decades and also uses modern filter and digi-tal signal processing theory to realize those concepts as a practical

and useful indicator. The RVI merges the old concepts with the new tech-nologies. The basic idea of the RVI is that prices tend to close higher than they open in up markets and tend to close lower than they open in down markets. The vigor of the move is thus established by where the prices reside at the end of the day. To normalize the index to the daily trading range, the change in price is divided by the maximum range of prices for the day. Thus, the basic equation for the RVI is

Close − OpenRVI = �� (6.1)

High − Low

In 1972, Jim Waters and Larry Williams published a description of their A/D Oscillator.1 In this case, A/D means accumulation/distribution rather than the usual advance/decline. Waters and Williams defined Buying Power (BP) and Selling Power (SP) as

BP = High − Open SP = Close − Low

where the prices were the open, high, low, and closing prices for the day.The two values, BP and SP, show the additional buying strength relative tothe open and the selling strength relative to the close to obtain an implied

55

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� � � � ��

��


measure of the day’s trading. Waters and Williams combined the measure-ment as the Daily Raw Figure (DRF). DRF is calculated as

BP + SPDRF = �� (6.2)

2 * (High − Low)

The maximum value of 1 is reached when a market opens trading at the low and closes at the high. Conversely, the minimum value of 0 is reached when the market opens trading at the high and closes at the low. The day-to-day evaluation causes the DRF to vary radically and requires smoothing to make it usable.

We can expand the equation for the DRF as

DRF =

=

=

1�2

1�2

1�2

�High − Open + Close − Low ��High − Low

High − Low + Close − OpenHigh − Low

Close − Open1 + (6.3)

High − Low

Clearly, the equation for the DRF is identical with the daily RVI expres-sion except for the additive and multiplicative constants. It seems there are no new ideas in technical analysis. However, smoothing must be done to make the indicator practical. This is where modern filter theory contributes to the successful implementation of the RVI. I use the four-bar symmetrical finite impulse response (FIR) filter (described in Equation 4.1 and Figure 4.1) to independently smooth the numerator and the denominator.

The RVI is an oscillator, and we are therefore only concerned with the cycle modes of the market in its use. The sharpest rate of change for a cycle is at its midpoint. Therefore, in the ascending part of the cycle we would expect the difference between the close and open to be at a maxi-mum. This is like a derivative in calculus, where the derivative of a sinewave produces a negative cosine wave. The derivative is therefore a waveform that leads the original sinewave by a quarter cycle. Also, from calculus, integration of a sinewave over a half-cycle period results in another sinewave delayed by a quarter cycle. Summing over a half cycle is basically the same as mathematically integrating, with the result that the waveshape of the sum is delayed by a quarter wavelength relative to the input. The net result of taking the differences and summing produces an oscillator output in phase with the cyclic component of the price. It is also possible to generate a leading function if the summation window is

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57 Relative Vigor Index

less than a half wavelength of the Dominant Cycle. If a cycle measure-ment is not available, you can sum the RVI components over a fixed default period. A nominal value of 8 is suggested because this is approxi-mately half the period of most cycles of interest.

Calculating the RVI is straightforward. The numerator, consisting of Close − Open, is filtered in the four-bar symmetrical FIR filter before the terms are summed. The denominator, consisting of High − Low, is indepen-dently filtered in the four-bar symmetrical FIR filter before it is summed. The numerator and denominator are summed individually and the RVI is then computed as the ratio of the numerator to the denominator. Since the numerator and denominator are lagged the same amount due to filtering, the lag is removed by taking their ratio.

The rules for the use of the RVI are flexible. Just remember that it is an oscillator that is basically in phase with the cyclic component of the mar-ket prices. I prefer crossing line indicators because they are unambiguous in their signals. A simple Trigger line is just the RVI delayed by one bar.

The RVI oscillator is shown in Figure 6.1. The responsiveness and clar-ity of the signals are self-explanatory. The EasyLanguage code to compute the RVI is shown in Figure 6.2, and its eSignal Formula Script (EFS) code is shown in Figure 6.3.

FIGURE 6.1 The RVI Gives Crisp Indications of the Cyclic Turning Point

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Inputs: Length(10);

Vars: Num(0), Denom(0), count(0), RVI(0), Trigger(0);

Value1 = ((Close – Open) + 2*(Close[1] - Open[1]) + 2*(Close[2] - Open[2]) + (Close[3] - Open[3]))/6;

Value2 = ((High – Low) + 2*(High[1] - Low[1]) + 2*(High[2] - Low[2]) + (High[3] - Low[3]))/6;

Num = 0;Denom = 0;For count = 0 to Length -1 begin

Num = Num + Value1[count];Denom = Denom + Value2[count];

End;If Denom <> 0 then RVI = Num / Denom;Trigger = RVI[1];

Plot1(RVI, “RVI”);Plot2(Trigger, “Trigger”);

FIGURE 6.2 EasyLanguage Code to Compute the RVI

/*****************************************************Title: RVI Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

======================================================

FIGURE 6.3 EFS Code to Compute the RVI

c06.qxd 2/2/04 10:45 AM Page 59


Fix History:


======================================================*****************************************************/

//External Variablesvar aRVIArrayvar aValue1Arrayvar aValue2Array

= new Array();= new Array();= new Array();



setPriceStudy(false);setStudyTitle(“RVI”);setCursorLabelName(“RVI”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );


aRVIArray[x] = 0.0; aValue1Array[x] = 0.0; aValue2Array[x] = 0.0; aValue3Array[x] = 0.0;

}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;

(continued)


c06.qxd 2/2/04 10:45 AM Page 60



OscLength = 8;}


}


aRVIArray.pop();aRVIArray.unshift( 0 );

aValue1Array.pop();aValue1Array.unshift( 0 );


}

aValue1Array[0] = ( ( close()-open() ) + 2*( close(-1)-open(-1) ) + 2*( close(-2)-open(-2) ) + ( close(-3)-open(-3) ) ) / 6;

aValue2Array[0] = ( ( high()-low() ) + 2*( high(-1)-low(-1) ) + 2*( high(-2)-low(-2) ) + ( high(-3)-low(-3) ) ) / 6;


for ( x=0; x<OscLength; x++ ){nNum += aValue1Array[x];nDenom += aValue2Array[x];


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}

if ( nDenom != 0 ) aRVIArray[0] = nNum/nDenom;

//return the calculated values{

return new Array( aRVIArray[0],_aRVIArray[1] );

}

}



• The RVI concept is that prices close higher than they open in up mar-kets and close lower than they open in down markets.

• The RVI is a normalized oscillator, where the movement is normalized to the trading range of each bar.

• Lag-canceling four-bar symmetrical FIR filters are used to produce a readable indicator.

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c07.qxd 2/2/04 10:46 AM Page 63

CHAPTER 7

Oscillator Comparison

“Let’s play musical chairs,” said Tom deceitfully.

In the previous three chapters I have described three different oscilla-tors using three different principles. There is probably no need for more than one oscillator in your technical trading arsenal if it is a good

one. It is my experience that a number of traders suffer from the “paralysis of analysis.” Rather than searching for the ideal combination of tools—or worse, changing the mix of tools for every situation—it is better to settle on the few tools that work the best for you on average. The three oscilla-tors are for your consideration. The only way to know which of the three is best is to do a comparison on the same chart using the same data for each. This comparison is shown in Figure 7.1.

Frankly, I don’t see a nickel’s worth of difference between the three oscillators in this particular example. All three indicate the relative cycle amplitude and correctly identify each major turning point as it occurs. If anything, the Relative Vigor Index (RVI) is slightly less susceptible to whip-saw indications. Nonetheless, I am partial to the Cyber Cycle because I know it contains only the theoretical cycle components that comprise an oscillator. I have seen greater differences between the oscillators in other data samples.

The differences will become more apparent when you insert these oscillators as part of an automatic trading strategy. In these applications one oscillator may give a signal one bar earlier than the others at critical times for the strategy. It’s also true that one oscillator may have fewer short-term crossovers that lead to whipsaw trades. In any event, you now have three excellent tools for your own technical analysis. It may be that one of the oscillators will outperform the others in your application.

63

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FIGURE 7.1 Comparison of the Cyber Cycle, CG, and RVI Oscillators

It may be constructive to compare just one of the oscillators I have developed to several other oscillators that are in common use on a chart using the same data as before. This standardized comparison is useful to assess the relative lag of the trading signals and the degree to which whip-saw signals are produced. Two of the more popular oscillators are the Relative Strength Index (RSI) and the Stochastic. These are compared to the Cyber Cycle in Figure 7.2, where eight-bar periods are used for compa-rable scaling. Whoa! Clearly, the RSI and Stochastic are more erratic than the Cyber Cycle. Waiting for confirmation for the indicators to cross the signal lines is the conventional way of minimizing the erratic behavior of the indicators. Waiting for confirmation means that the RSI and Stochastic trading signals are invariably late or that the signal is missed altogether. I could cite many more examples and many more comparison indicators, but the purpose of this book is to generate tools you can use in your own work. Since you have the code, you can test your own examples. You can also compare these new tools to your other favorite indicators.

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65 Oscillator Comparison

FIGURE 7.2 Cyber Cycle is Smoother and Signals are More Timely than Those of an RSI or Stochastic


• The Cyber Cycle, CG, and RVI oscillators all carry relative cycle ampli-tude information.

• The Cyber Cycle, CG, and RVI all indicate major turning points with minimum lag.

• The Cyber Cycle, CG, and RVI are vastly superior to standard indicators.

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c08.qxd 2/2/04 10:47 AM Page 67

CHAPTER 8

Stochasticization and Fisherization

of Indicators

“I’m of greater value to you every day,” said Tom appreciatively.

There is an indicator I wish I had invented because it works pretty well. This indicator is called the Stochastic RSI. Since I didn’t invent it, the best I can do is to describe it and then proceed to shamelessly

adapt some of its principles to create even better indicators. All of these indicators will be described and compared in this chapter.

The name of the Stochastic RSI is descriptive of how it is calculated. First an RSI Indicator is computed from recent prices; then a Stochastic Indicator is computed using the RSI as the input variable. Finally, a weighted moving average of the Stochastic is taken to smooth it so that a workable output can be viewed.

An RSI averages the difference in ascending closing prices over a selected period separately from averaging the difference in descending closing prices. For a shorthand notation, I will call these Closes Up (CU) and Closes Down (CD). The RSI is the ratio of CU to the sum of CU and CD. If there are no CD in the selected period, the ratio is unity (1). If there are no CU in the selected period, the ratio is 0. So, if the length exactly coin-cides with half the period of a perfect cycle, the RSI will swing between 0 and 1. It is common to multiply the ratio by 100 to display the RSI as a percentage.

A Stochastic Indicator1 (definitely not a stochastic random variable) is computed by finding the highest value and the lowest value over a selected period. The Stochastic Indicator is the ratio of the difference between the current and lowest values and the difference between the highest and low-est values. It is also common to multiply this ratio by 100 to display the Stochastic Indicator as a percentage.

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The EasyLanguage and EFS codes to derive the Stochastic RSI from RSI and Stochastics are given in Figures 8.1 and 8.2, respectively. Before the Stochastic RSI is plotted, it is smoothed by a weighted moving average to provide a pleasing display with minimal lag. The Trigger line displayed is the Signal line delayed by one bar. The crossing of the Stochastic RSI Signal and a Trigger line constitutes buy and sell signals of the indicator. I have taken the liberty of scaling the Stochastic RSI to swing between −1 and +1. My scaling is selected so that I can directly apply the Fisher transform to it to generate razor-sharp entry and exit signals.

The amazing thing about the Stochastic RSI is that, after all the com-putations, the trading signals have almost no lag. The primary reason for this is that both the RSI and the Stochastic Indicators are ratios, so that lag in the numerator is canceled by lag in the denominator. The performance of the Stochastic RSI is shown in Figure 8.3. Unlike the Cyber Cycle and CG Oscillator, the Stochastic RSI tends not to retain the relative amplitudes of the cycles. This amplitude standardization can be an advantage to traders because it removes some of the interpretive aspects of the oscillators. In Figure 8.3, the Stochastic RSI clearly captures every major turning point in a timely manner.

This is the good part. If taking the Stochastic of a standard indicator produces a better indicator, it is reasonable that a superlative indicator can be created by applying the same process to an already good indicator. The EasyLanguage and EFS codes for transforming the Cyber Cycle of Figure 4.4 into a Stochastic Cyber Cycle are given in Figures 8.4 and 8.5, respec-tively. The EasyLanguage and eSignal Formula Script (EFS) codes for

Inputs: RSILength(8),StocLength(8),WMALength(8);

Value1 = RSI(Close, RSILength) - Lowest(RSI(Close, RSILength), StocLength);

Value2 = Highest(RSI(Close, RSILength), StocLength) - Lowest(RSI(Close, RSILength), StocLength);

If Value2<> 0 then Value3 = Value1 / Value2;Value4 = 2*(WAverage(Value3, WMALength) - .5);

Plot1(Value4, “StocRSI”);Plot2(Value4[1], “Trig”);

FIGURE 8.1 EasyLanguage Code to Compute the Stochastic RSI

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69 Stochasticization and Fisherization of Indicators

/*****************************************************Title: Stochastic RSICoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/

//External Variables var nAvgUpClose = 0; var nAvgDnClose = 0; var ntAvgUpClose = 0; var ntAvgDnClose = 0; var nValue3 = 0; var nValue4 = 0; var nTrig = 0; var bInitialized = false;

var nRS = 0; var nRSI = 0;

var aRSIArray = new Array(); var aValue3Array = new Array();

//== PreMain function required by eSignal to set_ things up


setPriceStudy(false);setStudyTitle(“StochasticRSI”);

(continued)

FIGURE 8.2 EFS Code to Compute the Stochastic RSI

c08.qxd 2/2/04 10:47 AM Page 70


setCursorLabelName(“StocRSI”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );


aRSIArray[x] = 0.0;aValue3Array[x] = 0.0;

}

}

//== Main processing functionfunction main( RSILength, StocLength, WMALength ) {var x;var nDiff;var nDivBy;var nValue1;var nValue2;

//initialize parameters if necessaryif ( RSILength == null ) {

RSILength = 8;}if ( StocLength == null ) {

StocLength = 8;}if ( WMALength == null ) {

WMALength = 8;}


}

//initialize the basic RSI calculationif ( bInitialized == false ) {

nAvgUpClose = 0.0;nAvgDnClose = 0.0;for (x=0; x<RSILength; x++) {


c08.qxd 2/2/04 10:47 AM Page 71


nDiff = close( -x ) - close( -(x+1) );

if ( nDiff > 0 ) {nAvgUpClose += nDiff;

}else {

nAvgDnClose+= Math.abs

( nDiff );}

}nAvgUpClose /= RSILength;nAvgDnClose /= RSILength;nRS = nAvgUpClose / nAvgDnClose;nRSI = 100.0 - ( 100.0 / ( 1.0

+ nRS ) );

bInitialized = true;}//continue the RSI calculation on subsequent_

barselse {

if ( getBarState() == BARSTATE_NEWBAR ) {nAvgUpClose = ntAvgUpClose;nAvgDnClose = ntAvgDnClose;if ( !isNaN( nRSI ) ) {

aRSIArray.pop();aRSIArray.unshift( 0 );aValue3Array.pop();aValue3Array.unshift( 0 );nTrig = nValue4;

}}nDiff = close( 0 ) - close( -1 );if ( nDiff > 0 ) {

ntAvgUpClose = (( nAvgUpClose * (RSILength-1) ) + nDiff ) / RSILength;

ntAvgDnClose = (( nAvgDnClose * (RSILength-1) ) + 0 ) / RSILength;

}(continued)


c08.qxd 2/2/04 10:47 AM Page 72


else {ntAvgUpClose = (( nAvgUpClose

* (RSILength-1) ) + 0 ) / RSILength;

ntAvgDnClose = (( nAvgDnClose * (RSILength-1) ) + Math.abs( nDiff ) ) / RSILength;

}nRS = ntAvgUpClose / ntAvgDnClose;nRSI = 100.0 - ( 100.0 / ( 1

+ nRS ) );aRSIArray[0] = nRSI;

}

//calculate the StocRSI using the RSI Array we_ have created.

nValue1 = nRSI - Lowest( StocLength );nValue2 = Highest( StocLength )

- Lowest( StocLength );

nValue3 = 0;if ( nValue2 != 0 ) nValue3 = ( nValue1

/ nValue2 );aValue3Array[0] = nValue3;

//compute weighted moving averagenValue4 = 0;nDivBy = 0;for (x=0; x<WMALength; x++) {

nValue4 += ( aValue3Array[x] * ( WMALength-x ) );

nDivBy += ( WMALength-x );}

nValue4 = nValue4 / nDivBy;nValue4 = 2.0 * ( nValue4 - 0.5 );

//return the calculated valuesif (!isNaN( nValue4 ) ) {

return new Array( nValue4, nTrig );


c08.qxd 2/2/04 10:47 AM Page 73


}

}

/*****************************************************SUPPORT FUNCTIONS

*****************************************************/

function Highest( nPeriod ) {var x;var nTmp = -999999999.0;

for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aRSIArray[x] );

}

return( nTmp );}

function Lowest( nPeriod ) {var x;var nTmp = 999999999.0;

for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aRSIArray[x] );

}

return( nTmp );}


converting the CG Indicator of Figure 5.3 into a Stochastic CG are given in Figures 8.6 and 8.7, respectively. Finally, the EasyLanguage and EFS codes to stochasticize the Relative Vigor Index (RVI) of Figure 6.1 are provided in Figures 8.8 and 8.9, respectively. In each case, I have simply added the code to take the Stochastic of the indicators and scaled the resulting indicators to range between −1 and +1. This scaling was done because the next step of

c08.qxd 2/2/04 10:47 AM Page 74


FIGURE 8.3 The Stochastic RSI Captures Turning Points in a Timely Manner

the indicator creation is to take the Fisher transform for sharper, better-defined entry and exit signals. The Trigger is just the indicator delayed by one bar and scaled to swing between −0.98 and +0.98. Shrinking the size of the Trigger gives a better-defined crossover when the indicator moves away from the extreme values.

The three stochasticized indicators are compared in Figure 8.10. They are clearly similar, although I like the Stochastic Cyber Cycle because the buy/sell indications are based purely on the cycle content of the data. On the other hand, it is clear that the Stochastic RVI is more regular with fewer whipsaws. In any event, you have all the tools to make your own selection. The input parameters for each of the indicators enable you to optimize the indicator period, if desired, when used with a specific security.

In Chapter 1, I pointed out that the probability density functions of sinewaves are not Gaussian and that creating sharp indicators from them is difficult because the indications come after the movement has already started. The Stochasticized Indicators all look somewhat like sinewaves. Therefore, we should be able to create razor-sharp trading signals by apply-ing the Fisher transform to them. This is exactly what I have done in the indicator codes of Figures 8.11 through 8.16. I have limited the amplitude swings to absolute values of 0.99 to avoid getting huge output amplitudes from the Fisher transform. The trading signals, as before, are given by the crossing of the Signal line and the Trigger line. The Trigger line is simply the Signal line delayed by one bar.

c08.qxd 2/2/04 10:47 AM Page 75


{*****************************************************Stochastic Cyber Cycle

*****************************************************}

Inputs: Price((H+L)/2),alpha(.07),Len(8);

Vars: Smooth(0),Cycle(0),MaxCycle(0),MinCycle(0);




MaxCycle = Highest(Cycle, Len);MinCycle = Lowest(Cycle, Len);If MaxCycle <> MinCycle then Value1 = (Cycle

- MinCycle) / (MaxCycle - MinCycle);Value2 = (4*Value1 + 3*Value1[1] + 2*Value1[2]

+ Value1[3]) / 10;Value2 = 2*(Value2 - .5);

Plot1(Value2, “Cycle”);Plot2(.96*(Value2[1] + .02), “Trigger”);Plot3(0,”Ref”);

FIGURE 8.4 EasyLanguage Code to Compute the Stochastic Cyber Cycle

/*****************************************************Title: Stochastic Cyber Cycle Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003

(continued)

FIGURE 8.5 EFS Code to Compute the Stochastic Cyber Cycle

c08.qxd 2/2/04 10:47 AM Page 76


Version: 1.0.0

======================================================Fix History:


======================================================******************************************************/


var nBarCount = 0; var nValue2 = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array(); var aCycleArray = new Array(); var aValue1Array = new Array();



setPriceStudy(false);setStudyTitle(“StochasticCyberCycle”);setCursorLabelName(“Cycle”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );


aPriceArray[x] = 0.0;aSmoothArray[x] = 0.0;aCycleArray[x] = 0.0;aValue1Array[x] = 0.0;


c08.qxd 2/2/04 10:47 AM Page 77


}

}

//== Main processing functionfunction main( Alpha, OscLength ) {var x;var nPrice;var nMaxCycle;var nMinCycle;


Alpha = 0.07;}if ( OscLength == null ) {

OscLength = 8;}


}


nBarCount++;





(continued)


c08.qxd 2/2/04 10:47 AM Page 78


nTrig = nValue2;}





}else {

aCycleArray[0] = ( 1 - 0.5*Alpha ) * ( 1 - 0.5*Alpha ) * ( aSmoothArray[0] - 2*aSmoothArray[1] + aSmoothArray[2] ) + 2*( 1-Alpha ) * aCycleArray[1] - ( 1-Alpha ) * ( 1- Alpha ) * aCycleArray[2];

}

nMaxCycle = Highest( OscLength );nMinCycle = Lowest( OscLength );

if ( nMaxCycle != nMinCycle ) aValue1Array[0] = ( aCycleArray[0]-nMinCycle ) / ( nMaxCycle - nMinCycle );

nValue2 = ( 4*aValue1Array[0] + 3*aValue1Array[1] + 2*aValue1Array[2] + aValue1Array[3] ) / 10;

nValue2 = 2 * ( nValue2 - 0.5 );

if (!isNaN( nValue2 ) ) {return new Array( nValue2,

(0.96*(nTrig+0.02)) );}

}


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*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aCycleArray[x] );

}

return( nTmp );}


for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aCycleArray[x] );

}

return( nTmp );}


{*****************************************************Stochastic CG Oscillator

*****************************************************}

Inputs: Price((H+L)/2),Length(8);

Vars: count(0),Num(0),Denom(0),CG(0),

(continued)

FIGURE 8.6 EasyLanguage Code to Compute the Stochastic CG

c08.qxd 2/2/04 10:47 AM Page 80


MaxCG(0),MinCG(0);

Num = 0;Denom = 0;For count = 0 to Length - 1 begin



MaxCG = Highest(CG, Length);MinCG = Lowest(CG, Length);If MaxCG <> MinCG then Value1 = (CG - MinCG) / (MaxCG

- MinCG);Value2 = (4*Value1 + 3*Value1[1] + 2*Value1[2]


Plot1(Value2, “CG”);Plot2(.96*(Value2[1] + .02), “Trigger”);Plot3(0,”Ref”);


/*****************************************************Title: Stochastic CG Oscillator Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

======================================================Fix History:


FIGURE 8.7 EFS Code to Compute the Stochastic CG

c08.qxd 2/2/04 10:47 AM Page 81


======================================================*****************************************************/

//External Variablesvar nPrice = 0;var nCG = 0;var nValue2 = 0;var nTrig = 0;

var aPriceArrayvar aCGArrayvar aValue1Array




setPriceStudy(false);setStudyTitle(“StochasticCGOsc”);setCursorLabelName(“CG”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );


aPriceArray[x] = 0.0; aCGArray[x] = 0.0; aValue1Array[x] = 0.0;

}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;

(continued)


c08.qxd 2/2/04 10:47 AM Page 82


var nMaxCG;var nMinCG;var nValue1;


OscLength = 8;}

// study is initializingif (getBarState() == BARSTATE_ALLBARS) {return null;}





nTrig = nValue2;}



for ( x=0; x<OscLength; x++ ){nNum += ( 1.0 + x )

* ( aPriceArray[x] );nDenom += ( aPriceArray[x] );

}


c08.qxd 2/2/04 10:47 AM Page 83



aCGArray[0] = nCG;

nMaxCG = Highest( OscLength);nMinCG = Lowest( OscLength );

nValue1 = 0;if ( nMaxCG != nMinCG ) nValue1 = (nCG

- nMinCG) / (nMaxCG - nMinCG);aValue1Array[0] = nValue1;


nValue2 = 2.0 * ( nValue2 - 0.5 );

//return the calculated valuesif ( !isNaN( nValue2 ) ) {

return new Array( nValue2, (0.96*(nTrig+0.02)) );

}

}


*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aCGArray[x] );

}

return( nTmp );(continued)


c08.qxd 2/2/04 10:47 AM Page 84


}


for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aCGArray[x] );

}

return( nTmp );}


{****************************************************Stochastic Relative Vigor Index (RVI)

*****************************************************}Inputs: Length(8);

Vars: Num(0),Denom(0),count(0),RVI(0),MaxRVI(0),MinRVI(0);

Value1 = ((Close - Open) + 2*(Close[1] - Open[1]) + 2*(Close[2] - Open[2]) + (Close[3] - Open[3]))/6;

Value2 = ((High - Low) + 2*(High[1] - Low[1]) + 2*(High[2] - Low[2]) + (High[3] - Low[3]))/6;



End;

FIGURE 8.8 EasyLanguage Code to Compute the Stochastic RVI

c08.qxd 2/2/04 10:47 AM Page 85


If Denom <> 0 then RVI = Num / Denom;

MaxRVI = Highest(RVI, Length);MinRVI = Lowest(RVI, Length);If MaxRVI <> MinRVI then Value3 = (RVI - MinRVI)

/ (MaxRVI - MinRVI);Value4 = (4*Value3 + 3*Value3[1] + 2*Value3[2]


Plot1(Value4, “RVI”);Plot2(.96*(Value4[1] + .02), “Trigger”);Plot3(0,”Ref”);


/*****************************************************Title: Stochastic RVI Coded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/

//External Variables var nValue4 = 0; var nTrig = 0;

(continued)

FIGURE 8.9 EFS Code to Compute the Stochastic RVI

c08.qxd 2/2/04 10:47 AM Page 86


var aRVIArrayvar aValue1Arrayvar aValue2Arrayvar aValue3Array

= new Array();= new Array();= new Array();= new Array();



setPriceStudy(false);setStudyTitle(“StochasticRVI”);setCursorLabelName(“RVI”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );



}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;var nMaxRVI;var nMinRVI;


OscLength = 8;}


c08.qxd 2/2/04 10:47 AM Page 87


// study is initializingif (getBarState() == BARSTATE_ALLBARS) {

return null;}






nTrig = nValue4;}

aValue1Array[0] = ( ( close()-open() ) + 2*( close(-1)-open(-1) ) + 2*( close(-2)- open(-2) ) + ( close(-3)-open(-3) ) ) / 6;

aValue2Array[0] = ( ( high()-low() ) + 2*( high(-1)-low(-1) ) + 2*( high(-2)- low(-2) ) + ( high(-3)-low(-3) ) ) / 6;



}

if ( nDenom != 0 ) aRVIArray[0] = nNum/nDenom;(continued)


c08.qxd 2/2/04 10:47 AM Page 88


nMaxRVI = Highest( OscLength);nMinRVI = Lowest( OscLength );

if ( nMaxRVI != nMinRVI ) aValue3Array[0] = ( aRVIArray[0]-nMinRVI ) / ( nMaxRVI-nMinRVI );


nValue4 = 2.0 * ( nValue4 - 0.5 );


return new Array( nValue4, (0.96*(nTrig+0.02)) );

}

}/*****************************************************

SUPPORT FUNCTIONS*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aRVIArray[x] );

}

return( nTmp );}


for (x=0; x<nPeriod; x++) {


c08.qxd 2/2/04 10:47 AM Page 89


nTmp = Math.min( nTmp, aRVIArray[x] );}

return( nTmp );}


FIGURE 8.10 Comparison of the Stochasticized Indicators

{*****************************************************Fisher Cyber Cycle

*****************************************************}Inputs: Price((H+L)/2),

alpha(.07),Len(8);

Vars: Smooth(0),(continued)

FIGURE 8.11 EasyLanguage Code to Compute the Fisher Stochastic Cyber Cycle

c08.qxd 2/2/04 10:47 AM Page 90


Cycle(0),MaxCycle(0),MinCycle(0),Lead(0);


Cycle = (1 - .5*alpha)*(1 - .5*alpha)*(Smooth - 2*Smooth[1] + Smooth[2]) + 2*(1 - alpha)*Cycle[1]- (1 - alpha)*(1 - alpha)*Cycle[2];


MaxCycle = Highest(Cycle, Len);MinCycle = Lowest(Cycle, Len);If MaxCycle <> MinCycle then Value1 = (Cycle

- MinCycle) / (MaxCycle - MinCycle);Value2 = (4*Value1 + 3*Value1[1] + 2*Value1[2]

+ Value1[3]) / 10;

Value3 = .5*Log((1+1.98*(Value2-.5))/(1-1.98*(Value2-.5)));

Plot1(Value3, “Cycle”);Plot2(Value3[1], “Trigger”);Plot3(0,”Ref”);


/*****************************************************Title: Fisher Cyber CycleCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

FIGURE 8.12 EFS Code to Compute the Fisher Stochastic Cyber Cycle

c08.qxd 2/2/04 10:47 AM Page 91


======================================================Fix History:06/19/2003 - Initial Release1.0.0

======================================================*****************************************************/


var nBarCount = 0;var nValue3 = 0;

var aPriceArray = new Array();var aSmoothArray = new Array();var aCycleArray = new Array();var aValue1Array = new Array();

//== PreMain function required by eSignal to set things up


setPriceStudy(false);setStudyTitle(“FisherCyberCycle”);setCursorLabelName(“Cycle”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aValue1Array[x] = 0.0;

}

} (continued)


c08.qxd 2/2/04 10:47 AM Page 92


//== Main processing functionfunction main( Alpha, OscLength ) {var x;var nPrice;var nValue2;var nMaxCycle;var nMinCycle;


Alpha = 0.07;}if ( OscLength == null ) {

OscLength = 8;}


return null;}


nBarCount++;





nTrig = nValue3;}


c08.qxd 2/2/04 10:47 AM Page 93






}else {


}

nMaxCycle = Highest( OscLength );nMinCycle = Lowest( OscLength );

if ( nMaxCycle != nMinCycle ) aValue1Array[0] = ( aCycleArray[0]-nMinCycle ) / ( nMaxCycle - nMinCycle );


nValue3 = 0.5 * Math.log( ( 1 + 1.98 * ( nValue2-0.5 ) ) / ( 1 - 1.98 * ( nValue2-0.5 ) ) );

//return the calculated valuesif (!isNaN( nValue3 ) ) {

return new Array( nValue3, nTrig );}

}(continued)


c08.qxd 2/2/04 10:47 AM Page 94



*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aCycleArray[x] );

}

return( nTmp );}


for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aCycleArray[x] );

}

return( nTmp );}


c08.qxd 2/2/04 10:47 AM Page 95


{*****************************************************Fisher CG

*****************************************************}Inputs: Price((H+L)/2),

Length(8);

Vars: count(0),Num(0),Denom(0),CG(0),MaxCG(0),MinCG(0),Lead(0);




MaxCG = Highest(CG, Length);MinCG = Lowest(CG, Length);If MaxCG <> MinCG then Value1 = (CG - MinCG) /

(MaxCG - MinCG);Value2 = (4*Value1 + 3*Value1[1] + 2*Value1[2] +

Value1[3]) / 10;

Value3 = .5*Log((1+1.98*(Value2-.5))/(1-1.98*(Value2-.5)));

Plot1(Value3, "CG");Plot2(Value3[1], "Trigger");Plot3(0,"Ref");

FIGURE 8.13 EasyLanguage Code to Compute the Fisher Stochastic CG

c08.qxd 2/2/04 10:47 AM Page 96


/*****************************************************Title: Fisher Stochastic CG Oscillator Coded By: Chris D. Kryza (Divergence Software, Inc.) Email: [email protected] Incept: 06/19/2003 Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/

//External Variablesvar nPrice = 0;var nCG = 0;var nValue3 = 0;var nTrig = 0;

var aPriceArrayvar aCGArrayvar aValue1Array




setPriceStudy(false);setStudyTitle(“FisherStochasticCGOsc”);setCursorLabelName(“CG”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );

FIGURE 8.14 EFS Code to Compute the Fisher Stochastic CG

c08.qxd 2/2/04 10:47 AM Page 97



aPriceArray[x] = 0.0; aCGArray[x] = 0.0; aValue1Array[x] = 0.0;

}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;var nMaxCG;var nMinCG;var nValue1;


OscLength = 8;}


return null;}





(continued)


c08.qxd 2/2/04 10:47 AM Page 98


nTrig = nValue3;}



for ( x=0; x<OscLength; x++ ){nNum += ( 1.0 + x )


}


aCGArray[0] = nCG;

nMaxCG = Highest( OscLength);nMinCG = Lowest( OscLength );

if ( nMaxCG != nMinCG ) aValue1Array[0] = (nCG - nMinCG) / (nMaxCG - nMinCG);





}


c08.qxd 2/2/04 10:47 AM Page 99



*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aCGArray[x] );

}

return( nTmp );}


for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aCGArray[x] );

}

return( nTmp );}


{*****************************************************Fisher RVI

*****************************************************}

Inputs: Length(8);

Vars: Num(0),Denom(0),count(0),

(continued)

FIGURE 8.15 EasyLanguage Code to Compute the Fisher Stochastic RVI

c08.qxd 2/2/04 10:47 AM Page 100


RVI(0),Lead(0),MaxRVI(0),MinRVI(0);





End;If Denom <> 0 then RVI = Num / Denom;

MaxRVI = Highest(RVI, Length);MinRVI = Lowest(RVI, Length);If MaxRVI <> MinRVI then Value3 = (RVI - MinRVI)

/ (MaxRVI - MinRVI);Value4 = (4*Value3 + 3*Value3[1] + 2*Value3[2]

+ Value3[3]) / 10;

Value5 = .5*Log((1+1.98*(Value4 - .5))/(1-1.98*(Value4 - .5)));

Plot1(Value5, “RVI”);Plot2(Value5[1], “Trigger”);Plot3(0,”Ref”);


c08.qxd 2/2/04 10:47 AM Page 101


/*****************************************************Title: FisherStochastic RVICoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 06/19/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/

//External Variables var nValue5 = 0; var nTrig = 0;

var aRVIArray = new Array(); var aValue1Array = new Array(); var aValue2Array = new Array(); var aValue3Array = new Array();



setPriceStudy(false);setStudyTitle(“FisherStochasticRVI”);setCursorLabelName(“RVI”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );addBand( 0, PS_SOLID, Color.black, 1, -55 );


(continued)

FIGURE 8.16 EFS Code to Compute the Fisher Stochastic RVI

c08.qxd 2/2/04 10:47 AM Page 102



}

}

//== Main processing functionfunction main( OscLength ) {var x;var nNum;var nDenom;var nValue4;var nMaxRVI;var nMinRVI;


OscLength = 8;}


return null;}







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nTrig = nValue5;}

aValue1Array[0] = ( ( close()-open() ) + 2*( close(-1)-open(-1) ) + 2*( close(-2)-open(-2) ) + ( close(-3)-open(-3) ) ) / 6;

aValue2Array[0] = ( ( high()-low() ) + 2*( high(-1)-low(-1) ) + 2*( high(-2)-low(-2) )+ ( high(-3)-low(-3) ) ) / 6;



}

if ( nDenom != 0 ) aRVIArray[0] = nNum/nDenom;nMaxRVI = Highest( OscLength);nMinRVI = Lowest( OscLength );

if ( nMaxRVI != nMinRVI ) aValue3Array[0] = ( aRVIArray[0]-nMinRVI ) / ( nMaxRVI-nMinRVI );





}(continued)


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*****************************************************/


for (x=0; x<nPeriod; x++) {nTmp = Math.max( nTmp, aRVIArray[x] );

}

return( nTmp );}


for (x=0; x<nPeriod; x++) {nTmp = Math.min( nTmp, aRVIArray[x] );

}

return( nTmp );}


The three Fisherized indicators are compared in Figure 8.17. In all cases, the Fisher transform provides a means to filter the undesired whip-saw signals by ignoring line crossovers that happen at an absolute ampli-tude of less than 2. It appears that the Fisher RVI is the superior oscillator because, almost without exception, it provides trading signals several bars in advance of the other indicators. That makes it a really good indicator because the other two are not slouches in their own right. Any or all of the three can be a profound addition to your technical analysis tools.

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FIGURE 8.17 Fisherized Indicators Give Razor-Sharp Trading Signals


• New, easier-to-read oscillators can be created by applying the Stochastic calculation to existing indicators.

• The Stochastic RVI is an extraordinarily smooth and consistent oscil-lator.

• Performing a Fisher transform on amplitude-limited oscillators pro-vides a way to eliminate whipsaw signals by ignoring crossovers that occur at amplitudes less than 2.

• The Fisher RSI provides consistently timely signals with surgical pre-cision.

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c09.qxd 2/2/04 10:47 AM Page 107

CHAPTER 9

Measuring Cycles

“Looks like rain,” said Tom precipitously.

I t is obvious that cycles exist in the market. They can be found on any chart by the most casual observer. What is not so clear is how to iden-tify those cycles in real time and how to take advantage of their exis-

tence. When Welles Wilder first introduced the Relative Strength Index (RSI), I was curious as to why he selected 14 bars as the basis of his calcu-lations. I reasoned that if I knew the correct market conditions, then I could make indicators such as the RSI adaptive to those conditions. Cycles were the answer. I knew cycles could be measured. Once I had the cyclic measurement, a host of automatically adaptive indicators could follow.

Measurement of market cycles is not easy. The signal-to-noise ratio is often very low, making measurement difficult even using a good measure-ment technique. Additionally, the measurements theoretically involve simultaneously solving a triple infinity of parameter values. The parame-ters required for the general solutions were frequency, amplitude, and phase. Some standard engineering tools, like fast Fourier transforms (FFTs), are simply not appropriate for measuring market cycles because FFTs cannot simultaneously meet the stationarity constraints and produce results with reasonable resolution. Therefore I introduced Maximum Entropy Spectral Analysis (MESA) for the measurement of market cycles. This approach, originally developed to interpret seismographic informa-tion for oil exploration, produces high-resolution outputs with an excep-tionally short amount of information. A short data length improves the probability of having nearly stationary data. Stationary data means that fre-quency and amplitude are constant over the length of the data. I noticed over the years that the cycles were ephemeral. Their periods would be

107

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continuously increasing and decreasing. Their amplitudes also were chang-ing, giving variable signal-to-noise ratio conditions. Although all this is going on with the cyclic components, the enduring characteristic is that generally only one tradable cycle at a time is present for the data set being used. I prefer the term Dominant Cycle to denote that one component. The assumption that there is only one cycle in the data collapses the difficulty of the measurement process dramatically.

Assuming that only one cycle is present in the data enables the mea-surement to be made using a frequency discriminator. A frequency discrimi-nator basically measures the differential phase between successive samples. Since there are 360 degrees in each cycle, dividing 360 by the differential phase produces the measured cycle length. For example, if the differential phase is 20°, the resulting cycle length would be 360/20 = 18 bars. That is, an 18-bar cycle is changing phase at the rate of 20° per sample so that 360° (one cycle) is reached after 18 samples. Pretty simple! The most significant fact is that, in theory, the cycle measurement can be attained in just two samples.

To make the phase measurements, we need to describe the cycle in terms of a phasor instead of the conventional waveform with which we are familiar. The relationship between the cycle waveform and the phasor is shown in Figure 9.1. Imagine the phasor as the arrow whose tail is pinned at the origin and is rotating counterclockwise. A shadow cast by the arrow-head would then trace out the sinewave cycle. That is, as the phasor rotates, the peak amplitude is reached, followed by the zero crossing, fol-lowed by the minimum cycle amplitude, and then back to zero, and so on. One complete rotation of the phasor describes a cycle.

The phasor can be broken into two components, called the InPhase and Quadrature components, as shown in Figure 9.2. The phase angle for any given sample is easily found as the arctangent of the ratio of these two components.

FIGURE 9.1 A Phasor Can Represent a Cycle

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FIGURE 9.2 The Phase Angle Is the Arctangent of the Ratio of the Quadrature and InPhase Components

The trick is to break the analytic waveform (the cyclic component of prices in the form with which we are familiar) into the InPhase and Quadrature components. This is done with the Hilbert transform.1 The Hilbert transform is theoretically an infinite series; to make it practical for traders I have truncated the series at four elements. The equation for the Quadrature component in EasyLanguage notation is

Q = 0.0962 * Price + 0.5769 * Price[2] − 0.5769 * Price[4] − 0.0962 * Price[6]; (9.1)

The lag of the Quadrature component is half the filter length, or three bars. Therefore the InPhase component is just the price delayed by three bars, or

I = Price[3]; (9.2)

To test the speed of the cycle-measuring process, I created a single cycle of a 20-bar sinewave. I then applied the Hilbert transform, computed the phase angles, and used a discriminator to measure the cycle period. The results of this experiment are shown in Figure 9.3. These results are impressive. An accurate measurement of the cycle period is made within four samples of the beginning of the cycle. That four-sample lag is just the lag of the Hilbert transform plus one more sample because the phase dif-ference between samples is required for the computation of the period.

Before getting too excited about these results, please recall that this is a purely monochromatic theoretical waveform having an infinite signal-to-noise ratio. Furthermore, the waveform is already detrended because the cycle swings about the zero line. In the real world we must

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FIGURE 9.3 The Hilbert Transform Enables Rapid Measurement of the Cycle Period

detrend the signal to extract the cyclic component and then also deal with the noise that is superimposed on the signal. In other words, we need to compute the cyclic component of the market prices as we did in Figure 4.2 before we compute the cycle period.

The EasyLanguage and eSignal Formula Script (EFS) codes for com-puting the cycle period are shown in Figures 9.4 and 9.5, respectively. The description of the calculation is done with reference to Figure 9.4. After defining the inputs and declaring the variables, the first three lines of code recover the cyclic component, just as in Figure 4.2. The cyclic component is used to compute the Quadrature (Q1) and InPhase (I1) components of the Hilbert transform. One penalty for truncating the infinite series in com-puting the Quadrature component is that its amplitude is attenuated for the longer cycle periods. The last term in the computation of Q1 is a straight-line amplitude correction. Since the period is not yet known at this point in the code, and since the period is a relatively slowly varying function from sample to sample, it is satisfactory to use the period computed one bar ago in this compensation. I found this feedback compensation to be the most robust approach.

There is another amplitude compensation scheme that is possible. In the case of a pure cycle I can think of the InPhase component being Cos (θ)

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Inputs: Price((H+L)/2),alpha(.07);

Vars: Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),I2(0),Q2(0);




Q1 = (.0962*Cycle + .5769*Cycle[2] - .5769*Cycle[4] - .0962*Cycle[6])*(.5 + .08*InstPeriod[1]);

I1 = Cycle[3];

If Q1 <> 0 and Q1[1] <> 0 then DeltaPhase = (I1/Q1 - I1[1]/Q1[1]) / (1 + I1*I1[1]/(Q1*Q1[1]));

If DeltaPhase < 0.1 then DeltaPhase = 0.1;If DeltaPhase > 1.1 then DeltaPhase = 1.1;MedianDelta = Median(DeltaPhase, 5);

If MedianDelta = 0 then DC = 15 else DC = 6.28318 / MedianDelta + .5;

InstPeriod = .33*DC + .67*InstPeriod[1];Period = .15*InstPeriod + .85*Period[1];

Plot1(Period, “Period”);

FIGURE 9.4 EasyLanguage Code to Compute the Cycle Period

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/*****************************************************Title: Cycle Period Coded By: Chris D. Kryza (Divergence Software, Inc.) Email: [email protected] Incept: 06/19/2003 Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArrayvar aSmoothArrayvar aCycleArrayvar aDeltaPhasevar aPeriodvar aInstPeriodvar aQ1var aI1

= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();



setPriceStudy(false);setStudyTitle(“Cycle Period”);setCursorLabelName(“Period”, 0);setDefaultBarFgColor( Color.blue, 0 );

FIGURE 9.5 EFS Code to Compute the Cycle Period

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aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aQ1[x] = 0.0; aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha ) {var x;var nDC;var nMedianDelta;


Alpha = 0.07;}

// study is initializingif ( getBarState() == BARSTATE_ALLBARS ) {

return null;}


nBarCount++;



(continued)


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aQ1.pop();aQ1.unshift( 0 );

aI1.pop();aI1.unshift( 0 );

aDeltaPhase.pop();aDeltaPhase.unshift( 0 );

aInstPeriod.pop();aInstPeriod.unshift( 0 );

aPeriod.pop();aPeriod.unshift( 0 );

}

aPriceArray[0] = ( high()+low() ) / 2;




}else {


}


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aQ1[0] = ( 0.0962*aCycleArray[0] + 0.5769*aCycleArray[2] - 0.5769*aCycleArray[4] - 0.0962*aCycleArray[6] ) * ( 0.5 + 0.08 * aInstPeriod[1] );

aI1[0] = aCycleArray[3];

if ( aQ1[0] != 0 && aQ1[1] != 0 ) {aDeltaPhase[0] = (aI1[0]/aQ1[0]

- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));

}if ( aDeltaPhase[0] < 0.1 ) aDeltaPhase[0]

= 0.1;if ( aDeltaPhase[0] > 1.1 ) aDeltaPhase[0]

= 1.1;//Need a 5 bar Median filter of DeltaPhase here_

(MedianDelta)nMedianDelta = Median( 5, aDeltaPhase );

if ( nMedianDelta == 0 ) {nDC = 15;

}else {

nDC = 6.28318 / nMedianDelta + 0.5;}

aInstPeriod[0] = 0.33 * nDC + 0.67 * aInstPeriod[1];

aPeriod[0] = 0.15*aInstPeriod[0] + 0.85*aPeriod[1];

return( aPeriod[0] );

}

function Median( nBars, aArray ) {var aTmp = new Array();

(continued)


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var nTmp;var result;var x;

//transfer elements to temp arrayx = 0;while( x < nBars ) {

aTmp[x] = aArray[x++];}//sort array in asc orderaTmp.sort( SortAsc );

//if odd # of elements, just take middleif ( nBars % 2 != 0 ) {

result = aTmp[ (nBars+1) / 2 ]aTmp = null;return( result );

}//if even # elements, take average of two_

middle elementselse {

nTmp = nBars/2;result = (aTmp[nTmp] + aTmp[nTmp+1])/2;aTmp = null;return ( result );

}}

function SortAsc( arg1, arg2 ) {if (arg1<arg2) {

return( -1 )}else {

return( 1 );}

}


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� ��


and the Quadrature component being Sin (θ). Then, a compensation for amplitude error in the Quadrature component can be computed from the simple trigonometric identity

Sin2(θ) = 1 − Cos2(θ)

and normalizing amplitudes. While this is a great theory, and it works on the-oretical waveforms, I could not obtain satisfactory compensation on real price data because of the noise present in that data. I therefore use the feed-back amplitude compensation in the code.

The computation of the DeltaPhase starts with a conditional IF state-ment to preclude the possibility of dividing by 0. Some explanation for the rest of the line is required. The phase angle measured for the current bar is ArcTan (I1/Q1) and the phase angle for one bar ago is ArcTan (I1[1]/Q1[1]). The differential phase calculation is simplified using the trigonometric identity

A − BArcTan (A) − ArcTan(B) = ArcTan (9.3)

1 + AB

A six-bar cycle is as short as we need to measure. A six-bar cycle has a phase shift of 60° per bar, or 1.047 radians per bar. Since the differential phase has a maximum of about one radian, a reasonable approximation is that the angle in radians is approximately equal to the arctangent of that angle. This is the approximation we have applied to the computation of the differential angle in the code.

After the DeltaPhase is first computed, some limits must be estab-lished. First, the DeltaPhase must always be positive because time cannot run backward. If we get a negative DeltaPhase computation, it is either due to noise or because the two absolute phase measurements have split a quadrant of the phasor. (The arctangent is positive in quadrants 1 and 3 and is negative in quadrants 2 and 4.) In the case of a negative DeltaPhase, it is satisfactory to substitute the previous calculation. Instead, if the DeltaPhase is less than 0.1 radians I limit it to 0.1 radians. This is because a DeltaPhase smaller than 0.1 radians implies the period is greater than 63 bars (2 * π/0.1). The other limit is to not compute a period of less than six bars. This is done by limiting the DeltaPhase to 1.1 radians.

The actual calculation of the cycle period is perhaps the easiest part of the code to understand. In a nutshell, the concept is to divide the DeltaPhase into 2π because 2π represents one full cycle of phases in radian. measure. In practice, DeltaPhase is very noisy, varying by a large amount from bar to bar. If DeltaPhase were used directly, substantial smoothing would be required to recover a reasonable Dominant Cycle. There is a

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more efficient way of smoothing. The best kind of filter to use on spiky data is a median filter. Therefore I filter the DeltaPhases over five samples in a median filter to give the variable MedianDelta. MedianDelta is then divided into 2π to compute the Dominant Cycle. Measuring theoretical sinewave. periods, I found there is a bias of about 0.5 in the period measurement, and therefore added a compensation term to remove that bias. The Dominant Cycle is smoothed in an exponential moving average having α = 0.33 for a relatively rapid response for the feedback term in the computation of Q1. I call this variable the Instantanteous Period (InstPeriod). The InstPeriod is then smoothed again in an exponential moving average having α = 0.15. This value was selected to reach the full cycle length measurement in one cycle of a 20-bar signal, starting from 0.

I have conducted a number of rigorous tests to examine the quality of the cycle measurement. First among these is to examine the start-up tran-sient in a way similar to the single cycle measurement of Figure 9.3. The final results are shown in the bottom subgraph of Figure 9.6. In this case, I continue the 20-bar cycles after the first one. The InstPeriod comes up to a 20-bar measurement at 8 bars after initiation. This is consistent with the 1.5-bar lag of the smoothing filter plus the four-bar lag for the Hilbert

FIGURE 9.6 Measurement of a Single 20-Bar Cycle

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transform plus the 2.5-bar lag of the median filter. The smoothing of the period output is due to the exponential moving average. I could have used less smoothing. However, cycle periods tend to change relatively slowly in real data, and the greater amount of smoothing is desirable when lag is of less concern. These results should be viewed in context. For example, an FFT would take about 16 cycles of data to make a measurement of compa-rable resolution. Yes, you read it correctly—16 full cycles of data would be required by an FFT for equivalent results. Even MESA would take a large fraction of the cycle to make the first measurement.

With any measurement algorithm, one crucial test is whether the algo-rithm makes a correct measurement over a wide range of input data. To this end I created a theoretical sinewave whose period gradually increased from 6 bars to 40 bars. Figure 9.7 shows this waveform and shows that the measurement of its cycle periods is very accurate.

Another transient and accuracy test is to measure how fast the mea-surement algorithm can follow the switch from a 30-bar cycle to a 15-bar cycle and back. In Figure 9.8, the data consists of two cycles of a 30-bar

FIGURE 9.7 Measurement of a Chirped Waveform Whose Period Increases from 6 Bars to 40 Bars

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FIGURE 9.8 Measurement of Cycle Periods Varying from 30-Bar Cycles to 15-Bar Cycles and Back

cycle, four cycles of a 15-bar cycle, and two more cycles of a 30-bar cycle. This is a severe test, requiring the measurement to slew over a wide range between harmonically related cycles. This test shows that the measure-ment is within reasonable range of the actual period within 15 samples, switching either way.

The basic message here is that the cycle measurement has a lag of about 8 bars, as demonstrated in Figure 9.6, up to a lag of about 15 bars in one of the most stressing situations. This lag should be recognized when the measurement is used in trading.

Figure 9.9 shows the cycle period measurement of real data. This mea-surement is far more responsive than the more common measurements. Measurement accuracy can be tested by counting bars between major suc-cessive lowest lows or major successive highest highs and comparing the count to the measurement at that point. There are five bars per horizontal unit as a tip to help speed up your bar count. Please recall that there is about an eight-bar lag in the cycle measurement waveform.

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FIGURE 9.9 Cycle Measurement of Real Data


• The Hilbert transform enables the cycle period to be measured in as few as four bars.

• The cyclic component must be extracted from the data and then used to measure the Dominant Cycle period.

• The frequency discriminator to measure the Dominant Cycle period just sums the differential phases between bars until the sum reaches 360°—a full cycle.

• A five-bar median filter creates the differential phase to be summed. • Summing the median differential phase enables the cycle measure-

ment to be made using only five samples. • The lag of measuring the Dominant Cycle period is about eight bars. • The Dominant Cycle period measurement technique described in this

chapter is the most responsive technique available.

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CHAPTER 10

Adaptive Cycle Indicators

“The dinosaurs did not survive,” said Tom adaptively.

Having made the cycle period measurements as in Chapter 9, one brute force application would be to note the most recent highest high and then count forward the number of bars equal to half the

dominant cycle period to locate the next buying opportunity. Fortunately, we can be much more sophisticated in our analysis using indicators. If indi-cators work moderately well using fixed lengths in their computation, then these indicators should sparkle when the length is adaptive to a fraction of the measured dominant cycle.

I developed several oscillator-type indicators in Chapters 4 through 6. I will now revisit each of these and examine the improvements that result from using a Dominant Cycle measurement to make their computational length adaptive to the current market conditions. In each case, I compare the adaptive version of the indicator to the static version. I also compare the three adaptive indicators to each other for you to judge which is prefer-able. Since I use the same price chart throughout this book for consistency, and because you can test these indicators on your own computer using your own data, I will not bore you with agonizing details regarding indica-tor performance and comparisons.

ADAPTIVE CYBER CYCLE

The most simple cycle indicator was the Cyber Cycle, which was extractedfrom the price series in Chapter 4 by filtering out the trend component. The filter itself was derived in Chapter 2. This filter used the coefficient

123

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α = 0.07. The EasyLanguage and eSignal Formula Script (EFS) codes for the adaptive version of the Cyber Cycle Indicator are shown in Figures 10.1 and 10.2, respectively. Here, the Dominant Cycle is computed exactly as in Chapter 9. A fixed value of alpha is used to make the Dominant Cycle period measurement; then the measured Dominant Cycle is used to com-pute the coefficient alpha1. It is commonly recognized that the exponential moving alpha is related to the length of a simple moving average by the equation α = 2/(Length + 1). In this case, I use the Dominant Cycle period as the length in the computation of alpha1. This enables the Cyber Cycle Indicator to be adaptive to the measured Dominant Cycle period. A trigger signal consisting of the adaptive cycle delayed by one bar is also included in the indicator. Crossings of the adaptive cycle indicator and the trigger signal represent the buy and sell opportunities identified by this indicator.

Figure 10.3 shows the Adaptive Cyber Cycle Indicator compared to the static Cyber Cycle. This comparison shows that the adaptive indicator gen-erally emphasizes the cyclic swings and is often one bar earlier in produc-ing buy and sell signals.

ADAPTIVE CG INDICATOR

The CG Oscillator, derived in Chapter 5, finds the center of gravity of a fixed-length data sample as the sampling window is moved from bar to bar. The Adaptive CG Indicator uses half the measured Dominant Cycle period as the adaptive length of this variant of the CG Oscillator. The EasyLanguage and EFS codes for the adaptive version of the CG Oscillator are shown in Figures 10.4 and 10.5, respectively. Here, the dominant cycle is computed exactly as in Chapter 9. A fixed value of alpha is used to make the dominant cycle period measurement. The variable IntPeriod is com-puted as the integer portion of a four-bar weighted moving average of the Period. Since the weighted coefficients are divided by twice their sum, IntPeriod is the integer value of half the Dominant Cycle period. An integer value is required to sum the numerator and denominator in the subsequent code. Since the length of the summing varies with the length of the mea-sured Dominant Cycle period, the CG is adaptive to it.

Figure 10.6 shows the Adaptive CG Indicator compared to the static CG Oscillator. This comparison in this data set does not display any dra-matic change in the indicator as a result of making it adaptive.

ADAPTIVE RELATIVE VIGOR INDEX

The RVI, derived in Chapter 6, finds the difference of the close minus the open, normalized to the difference of the high and low. This ratio was

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125 Adaptive Cycle Indicators

{*****************************************************Adaptive Cycle

*****************************************************}Inputs: Price((H+L)/2),

alpha(.07);

Vars: Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),Length(0),Num(0),Denom(0),alpha1(0),AdaptCycle(0);





I1 = Cycle[3];




InstPeriod = .33*DC + .67*InstPeriod[1];(continued)

FIGURE 10.1 EasyLanguage Code for the Adaptive Cyber Cycle

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Period = .15*InstPeriod + .85*Period[1];

alpha1 = 2 / (Period + 1);AdaptCycle = (1 - .5*alpha1)*(1 - .5*alpha1)*(Smooth

- 2*Smooth[1] + Smooth[2]) + 2*(1 - alpha1)*AdaptCycle[1] - (1 - alpha1)*(1 - alpha1)*AdaptCycle[2];

If currentbar < 7 then AdaptCycle = (Price - 2*Price[1] + Price[2]) / 4;

Plot1(AdaptCycle, “AdaptCycle”);Plot2(AdaptCycle[1], “Trigger”);


/*****************************************************Title: Adaptive Cyber Cycle IndicatorCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array();

FIGURE 10.2 EFS Code for the Adaptive Cyber Cycle

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var aCycleArrayvar aDeltaPhasevar aPeriodvar aInstPeriodvar aQ1var aI1var aACycleArray

= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();



setPriceStudy(false);setStudyTitle(“Adaptive CyberCycle”);setCursorLabelName(“Cycle”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aQ1[x] = 0.0; aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0; aACycleArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha ) {var x;var Alpha1;var nDC;var nMedianDelta;

//initialize parameters if necessary(continued)


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if ( Alpha == null ) {Alpha = 0.07;

}


return null;}


nBarCount++;









aACycleArray.pop();aACycleArray.unshift( 0 );


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}





}else {


}




- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));



= 1.1;

nMedianDelta = Median( 5, aDeltaPhase );(continued)


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}else {




Alpha1 = 2 / ( aPeriod[0] + 1 );

if ( nBarCount < 7 ) {aACycleArray[0] = (aPriceArray[0]

- 2*aPriceArray[1] + aPriceArray[2])/4;

}else {

aACycleArray[0] = ( 1 - 0.5*Alpha1 ) * ( 1 - 0.5*Alpha1 ) * ( aSmoothArray[0] - 2*aSmoothArray[1] +

aSmoothArray[2] ) + 2*( 1 - Alpha1 ) * aACycleArray[1] - ( 1-Alpha1 ) * ( 1-Alpha1 ) * aACycleArray[2];

}

//return the calculated valuesif (!isNaN( aACycleArray[0] ) ) {

return new Array( aACycleArray[0],_ aACycleArray[1] );

}

}


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function Median( nBars, aArray ) {var aTmp = new Array();var nTmp;var result;var x;





}//if even # elements, take average of two middle

elementselse {


}}

function SortAsc( arg1, arg2 ) {if (arg1<arg2) {return( -1 )

}else {return( 1 );

}}


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FIGURE 10.3 Adaptive Cyber Cycle Indicator Is More Responsive to Cyclic Price Variations than Static Cyber Cycle Indicator

{*****************************************************Adaptive CG

*****************************************************}Inputs: Price((H+L)/2),

Vars:

alpha(.07);

Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),count(0),Num(0),Denom(0),CG(0),IntPeriod(0);

FIGURE 10.4 EasyLanguage Code to Compute the Adaptive CG Indicator

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I1 = Cycle[3];




InstPeriod = .33*DC + .67*InstPeriod[1];Value1 = .15*InstPeriod + .85*Value1[1];IntPeriod = intportion(Value1 / 2);

Num = 0;Denom = 0;For count = 0 to IntPeriod - 1 begin


End;If Denom <> 0 then CG = -Num/Denom + (IntPeriod + 1)

/ 2;

Plot1(CG, “CG”);Plot2(CG[1], “Trigger”);


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/*****************************************************Title: Adaptive CG OscillatorCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array(); var aCycleArray = new Array(); var aDeltaPhase = new Array(); var aPeriod = new Array(); var aInstPeriod = new Array(); var aQ1 = new Array(); var aI1 = new Array(); var aCGArray = new Array();



setPriceStudy(false);setStudyTitle(“Adaptive CG”);setCursorLabelName(“CG”, 0);

FIGURE 10.5 EFS Code to Compute the Adaptive CG Indicator

c10.qxd 2/2/04 10:50 AM Page 135


setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aQ1[x] = 0.0; aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0; aCGArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha ) {var x;var nCG = 0;var nDC;var nIntPeriod;var nNum;var nDenom;var nMedianDelta;


Alpha = 0.07;}


}


(continued)


c10.qxd 2/2/04 10:50 AM Page 136


nBarCount++;










}






c10.qxd 2/2/04 10:50 AM Page 137


}else {

aCycleArray[0] = ( 1 - 0.5*Alpha ) * ( 1 - 0.5*Alpha ) * ( aSmoothArray[0] - 2*aSmoothArray[1]+ aSmoothArray[2] ) + 2*( 1-Alpha ) * aCycleArray[1] - ( 1-Alpha ) * ( 1-Alpha ) * aCycleArray[2];

}




- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));



= 1.1;

nMedianDelta = Median( 5, aDeltaPhase );


}else {




(continued)


c10.qxd 2/2/04 10:50 AM Page 138


nIntPeriod = Math.floor( ( 4*aPeriod[0] + 3*aPeriod[1] + 2*aPeriod[3] + aPeriod[4] ) / 20 );


for ( x=0; x<nIntPeriod; x++ ){nNum += ( 1.0 + x )


}

if ( nDenom != 0 ) nCG = -nNum/nDenom + ( nIntPeriod+1 )/2;

aCGArray[0] = nCG;

//return the calculated valuesif (!isNaN( aCGArray[0] ) ) {

return new Array( aCGArray[0], aCGArray[1] );

}

}



aTmp[x] = aArray[x++];


c10.qxd 2/2/04 10:50 AM Page 139


}//sort array in asc orderaTmp.sort( SortAsc );



}//if even # elements, take average of two middle_

elementselse {


}}


return( -1 )}else {

return( 1 );}

}


computed over a fixed period. The Adaptive RVI Indicator uses half the measured Dominant Cycle period as the adaptive length of this variant of the RVI. The EasyLanguage and EFS codes for the adaptive version of the RVI are shown in Figures 10.7 and 10.8, respectively. Here the Dominant Cycle is computed exactly as in Chapter 9. A fixed value of alpha is used to make the Dominant Cycle period measurement. The variable Length is computed as the integer portion of a four-bar weighted moving average of the period. Since the weighted coefficients are divided by twice their sum,

c10.qxd 2/2/04 10:50 AM Page 140


FIGURE 10.6 Adaptive CG Indicator Compared to Static CG Oscillator

{*****************************************************Adaptive RVI

*****************************************************}Inputs: Price((H+L)/2),

Vars:

alpha(.07);

Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),count(0),Length(0),Num(0),Denom(0),RVI(0),MaxRVI(0),

FIGURE 10.7 EasyLanguage Code to Compute the Adaptive RVI

c10.qxd 2/2/04 10:50 AM Page 141


MinRVI(0);Smooth = (Price + 2*Price[1] + 2*Price[2]

+ Price[3])/6;Cycle = (1 - .5*alpha)*(1 - .5*alpha)*(Smooth

- 2*Smooth[1] + Smooth[2]) + 2*(1 - alpha)*Cycle[1]- (1 - alpha)*(1 - alpha)*Cycle[2];



I1 = Cycle[3];




InstPeriod = .33*DC + .67*InstPeriod[1];Period = .15*InstPeriod + .85*Period[1];Length = intportion((4*Period + 3*Period[1]

+ 2*Period[3] + Period[4]) / 20);





End;If Denom <> 0 then RVI = Num / Denom;

Plot1(RVI, “RVI”);Plot2(RVI[1], “Trigger”);


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/*****************************************************Title: Adaptive RVICoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array(); var aCycleArray = new Array(); var aDeltaPhase = new Array(); var aPeriod = new Array(); var aInstPeriod = new Array(); var aQ1 = new Array(); var aI1 = new Array(); var aRVIArray = new Array(); var aV1Array = new Array(); var aV2Array = new Array();



setPriceStudy(false);setStudyTitle(“Adaptive RVI”);

FIGURE 10.8 EFS Code to Compute the Adaptive RVI

c10.qxd 2/2/04 10:50 AM Page 143


setCursorLabelName(“RVI”, 0);setCursorLabelName(“Trig”, 1);setDefaultBarFgColor( Color.blue, 0 );setDefaultBarFgColor( Color.red, 1 );


aPriceArray[x] = 0.0;aSmoothArray[x] = 0.0;aCycleArray[x] = 0.0;aQ1[x] = 0.0;aI1[x] = 0.0;aDeltaPhase[x] = 0.0;aPeriod[x] = 0.0;aInstPeriod[x] = 0.0;aRVIArray[x] = 0.0;aV1Array[x] = 0.0;aV2Array[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha ) {var x;var nRVI = 0;var nDC;var nLength;var nNum;var nDenom;var nMedianDelta;


Alpha = 0.07;}


}

(continued)


c10.qxd 2/2/04 10:50 AM Page 144



nBarCount++;










aV1Array.pop();aV1Array.unshift( 0 );


}



c10.qxd 2/2/04 10:50 AM Page 145





}else {

aCycleArray[0] = ( 1 - 0.5*Alpha ) * ( 1 - 0.5*Alpha ) * ( aSmoothArray[0] - 2*aSmoothArray[1]+ aSmoothArray[2] ) + 2*( 1-Alpha ) * aCycleArray[1] - ( 1-Alpha ) * ( 1-Alpha ) * aCycleArray[2];

}




- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));



= 1.1;


nPhaseSum = 0; nOldPhaseSum = 0; nDC = 0;

if ( nMedianDelta == 0 ) {(continued)


c10.qxd 2/2/04 10:50 AM Page 146


nDC = 15;}else {




nLength = Math.floor( ( 4*aPeriod[0] + 3*aPeriod[1] +

2*aPeriod[3] + aPeriod[4] ) / 20 );

aV1Array[0] = ( ( close()-open() ) + 2*( close(-1)-open(-1) ) + 2*( close(-2)-open(-2) ) + ( close(-3)-open(-3) ) ) / 6;

aV2Array[0] = ( ( high()-low() ) + 2*( high(-1)-low(-1) ) + 2*( high(-2)-low(-2) ) + ( high(-3)-low(-3) ) ) / 6;


for ( x=0; x<nLength; x++ ){nNum += aV1Array[x];nDenom += aV2Array[x];

}

if ( nDenom != 0 ) nRVI = nNum/nDenom;aRVIArray[0] = nRVI;

//return the calculated valuesif (!isNaN( aRVIArray[0] ) ) {

return new Array( aRVIArray[0],_ aRVIArray[1] );

}


c10.qxd 2/2/04 10:50 AM Page 147


}







elementselse {


}}


return( -1 )}else {

return( 1 );}

}


c10.qxd 2/2/04 10:50 AM Page 148


FIGURE 10.9 The Adaptive RVI Is More Responsive to Shorter Cycle Variations than the Static RVI

FIGURE 10.10 Adaptive Indicator Comparison

c10.qxd 2/2/04 10:50 AM Page 149


Length is the integer value of half the Dominant Cycle period. An integer value is required to sum the numerator and denominator in the subsequent code. Since the length of the summing varies with the length of the mea-sured Dominant Cycle period, the RVI is adaptive to it.

Figure 10.9 shows the Adaptive RVI compared to the static RVI. As with the other adaptive indicators, the contributions of the shorter cycle periods are emphasized when compared to their static variants.

Figure 10.10 shows the three adaptive indicators compared to each other. As with their static variants, all show about the same performance. Perhaps the message is that once the cyclic component is extracted from the data correctly, most oscillator-type indicators have about the same per-formance. It may be true that one indicator performs better in one data set than another. The bottom line is that you now have in your toolbox three independently derived indicators from which to choose. It then becomes a matter of personal preference.


• The adaptive indicators all use the measured Dominant Cycle as their adaptive criterion.

• The Adaptive Cyber Cycle adapts to the full Dominant Cycle period in the computation of its alpha1 filter parameter.

• The Adaptive CG Oscillator uses the integer portion of the half Dominant Cycle period in the computation of the filter center of gravity.

• The Adaptive RVI uses the integer portion of the half Dominant Cycle period in the computation of the vigor ratio.

• All three adaptive indicators demonstrate similar performance.

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c11.qxd 2/2/04 10:58 AM Page 151

CHAPTER 11

The Sinewave Indicator

“I can forecast the future,” said Tom predictably.

Causal filters can never predict the future. In fact, all have lag. The purpose of making good indicators adaptive in Chapter 9 was to eliminate as much lag as possible, not to make a prediction. With the

Sinewave Indicator we are trying to create a noncausal filter that can pre-dict the turning point of market cycles. Anticipation of the cyclic turning points is a major advantage of the Sinewave Indicator when compared to other oscillators, such as the RSI and Stochastic Indicators, that must wait for confirmation.

In Chapter 9 I showed you how to measure the period of the dominant market cycle for any bar in the data series. However, this measurement does not tell us where we are within that cycle. To locate the position of the cycle, we must measure the phase of the Dominant Cycle. Knowing the phase of the cycle, we can take the sine of the measured phase to create an artificial oscillator-type indicator. That is, the cyclic component of the market data is synthesized as a pure sinewave. Any lag we created in the process of mea-suring the phase can be mathematically removed. Furthermore, simply adding 45° to the measured phase creates an artificial phase lead. This is the noncausal factor. The phase is advanced on the presumption that the mea-sured cycle has existed (at least briefly) in the past and will continue (at least briefly) into the future. Advancing the phase by 45° and taking a sine of the advanced phase angle produces an oscillator waveshape that leads the original sinewave by one-eighth of a cycle. The two sinewaves therefore cross 1⁄16 of a cycle before the peak cycle turning point and before the valley turning point. For a 16-bar Dominant Cycle, this gives an ideal 1-bar advance warning of the absolute Dominant Cycle turning points. For a 48-bar cycle,

151

c11.qxd 2/2/04 10:58 AM Page 152


the advance extends to 3 bars. For an eight-bar Dominant Cycle, the advance warning is theoretically only half a bar.

My simplified model of the market consists of a trend and a cycle. There are certainly additional components present in real markets, but we are ignor-ing them in this simplified model. I call the highest-amplitude cycle the Dominant Cycle. Experience bears out that the assumption of the presence of a single Dominant Cycle is a workable approximation. Knowing the Dominant Cycle period, the phase of this Dominant Cycle can be measured. But if the market goes into a pure trend, there is no cycle. In this case, the phase ceases to advance. If the phase does not advance, then the two sinewave waveshapes of the Sinewave Indicator cannot cross. If the two waveshapes do not cross, the Sinewave Indicator produces no cyclic buy or sell signals. This avoidance of false whipsaw signals is a distinct advantage over traditional oscillators. In practice, the phase does not exactly stop; the phase does languish and the phase waveshape appears distinctly different than the constant rate of change that is produced when the market is in a cycle mode. The phase varies between 0° and 360°. If the cycle period is changing, there is an occasional crossing of the Sinewave Indicator lines to correct the phase angle for the cur-rent cycle period measurement. In these cases, the Sinewave Indicator lines do not appear to be sinewaves in the vicinity of the crossing. Therefore, these occasional bad crossing signals are easy to identify.

We obtain the Sinewave Indicator by plotting the sine of the measured phase angle. This gives us an oscillator that always swings between the lim-its of −1 and +1. We enhance the usability of this oscillator by plotting the sine of the phase angle advanced by 45°. The effect of plotting these two lines is shown for both the phasor and time domain presentations in Figure 11.1. Adding 45° clearly advances the phasor from a 45° slant to the vertical position. This phase advance means the LeadSine waveform will crest before the Sine crests. The LeadSine and Sine lines cross 22.5°, or 1⁄ of a16

cycle, before the turning point of the cycle is reached. If the market has a cycle of 16 bars or less, this is a signal to enter or exit a trade immediately. If the market has a longer cycle, there is some built-in anticipation time before you pull the trigger.

FIGURE 11.1 Phasor and Time Domain Views of the Sinewave Indicator

c11.qxd 2/2/04 10:58 AM Page 153

153 The Sinewave Indicator

Compared to conventional oscillators such as the Stochastic or RSI, the Sinewave Indicator has two major advantages. These are as follows:

1. The Sinewave Indicator anticipates the Cycle Mode turning point rather than waiting for confirmation.

2. The phase does not advance when the market is in a Trend Mode. Therefore the Sinewave Indicator does not tend to give false whipsaw signals when the market is in a Trend Mode.

An additional advantage is that the anticipation signal is obtained strictly by mathematically advancing the phase. Momentum is not em-ployed. Therefore, the Sinewave Indicator signals are no more noisy than the original signal.

The EasyLanguage and eSignal Formula Script (EFS) codes to mea-sure Dominant Cycle phase and then to synthesize the Sinewave are described with reference to Figures 11.2 and 11.3, respectively. The initial part of the code measures the Dominant Cycle exactly as in Chapter 9. The measured period must be further smoothed using an exponential moving average (α = 0.15) because there is no further smoothing in the computa-tion of the phase. The variable DCPeriod is the integer portion of the smoothed Dominant Cycle period because it is used to sum over the period and only an integer variable can be used for this purpose. Otherwise, rounding errors cause erratic results. The cycle component of the data is multiplied individually with the sine and cosine of the Dominant Cycle period, and these two products are summed individually over one com-plete cycle. These sums are known as the real part and the imaginary part of the data. It is well known that the arctangent of their ratio is the phase of cycle component. The arctangent function can go to infinity, and the code precludes a computational problem if the ImagPart variable is smaller than 0.001. The arctangent function is also subject to ambiguities, depending on in which phase quadrant the computation resides. In EasyLanguage, it is simplest to resolve these ambiguities by rotating the DCPhase by 90° and then adding another 180° if ImagPart is negative. If converting this code to another language, care should be taken when dealing with the arctangent function. First, most computer languages represent angles in terms of radi-ans rather than degrees. Second, the ambiguity resolution scheme I used is not universally appropriate for all languages. The Sine Indicator is plotted simply as the sine of the phase angle of the Dominant Cycle and the LeadSine Indicator is plotted as the sine of the phase angle plus 45°, giving it the desired leading property.

The Sinewave Indicators are plotted against both theoretical analytic waveforms and real-world data to demonstrate their performance. Figure 11.4 shows a theoretical 20-bar cycle sinewave analytic waveform. Note

c11.qxd 2/2/04 10:58 AM Page 154


{*****************************************************Sinewave Indicator

*****************************************************}

Inputs: Price((H+L)/2),alpha(.07);

Vars: Smooth(0),Cycle(0),I1(0),Q1(0),I2(0),Q2(0),DeltaPhase(0),MedianDelta(0),MaxAmp(0),AmpFix(0),Re(0),Im(0),DC(0),alpha1(0),InstPeriod(0),DCPeriod(0),count(0),SmoothCycle(0),RealPart(0),ImagPart(0),DCPhase(0);




{Cycle = Price;}Q1 = (.0962*Cycle + .5769*Cycle[2] - .5769*Cycle[4]

- .0962*Cycle[6])*(.5 + .08*InstPeriod[1]);I1 = Cycle[3];


FIGURE 11.2 EasyLanguage Code to Compute the Sinewave Indicator

c11.qxd 2/2/04 10:58 AM Page 155




InstPeriod = .33*DC + .67*InstPeriod[1];Value1 = .15*InstPeriod + .85*Value1[1];

{Compute Dominant Cycle Phase}DCPeriod = IntPortion(Value1);RealPart = 0;ImagPart = 0;For count = 0 To DCPeriod - 1 begin

RealPart = RealPart + Sine(360 * count / DCPeriod) * (Cycle[count]);

ImagPart = ImagPart + Cosine(360 * count / DCPeriod) * (Cycle[count]);

End;If AbsValue(ImagPart) > 0.001 then DCPhase

= Arctangent(RealPart / ImagPart);If AbsValue(ImagPart) <= 0.001 then DCPhase = 90

* Sign(RealPart);

DCPhase = DCPhase + 90;If ImagPart < 0 then DCPhase = DCPhase + 180;If DCPhase > 315 then DCPhase = DCPhase - 360;

Plot1(Sine(DCPhase), “Sine”);Plot2(Sine(DCPhase + 45), “LeadSine”);


how the LeadSine crosses over the Sine immediately prior to each peak and valley in the price waveform. The LeadSine always crosses the Sine line before the turning point in the cycle, giving advance indication of the cyclic turning point. The amount of advance warning relative to the length of the cycle is less for the shorter cycles.

The Sinewave Indicator is plotted in the bottom graph for the standard data set in Figure 11.5. The market is in a trend at the left side of the chart in August and September. We know this because the wiggles in the

c11.qxd 2/2/04 10:58 AM Page 156


/*****************************************************Title: Sine Wave IndicatorCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array(); var aCycleArray = new Array(); var aDeltaPhase = new Array(); var aPeriod = new Array(); var aInstPeriod = new Array(); var aQ1 = new Array(); var aI1 = new Array(); var aV1Array = new Array();



setPriceStudy(false);setStudyTitle(“Sine Wave”);setCursorLabelName(“Sine”, 0);setCursorLabelName(“LeadSine”, 1);setDefaultBarFgColor( Color.blue, 0 );

FIGURE 11.3 EFS Code for the Sinewave Indicator

c11.qxd 2/2/04 10:58 AM Page 157


setDefaultBarFgColor( Color.red, 1 );


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aQ1[x] = 0.0; aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0; aV1Array[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha ) {var x;var nDC;var nDCPeriod;var nRealPart;var nImagPart;var nDCPhase = 0.0;var nMedianDelta;


Alpha = 0.07;}


}


nBarCount++;

(continued)


c11.qxd 2/2/04 10:58 AM Page 158











}





}else {


c11.qxd 2/2/04 10:58 AM Page 159



}




- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));



= 1.1;



}else {




aV1Array[0] = 0.15*aPeriod[0] + 0.85*aV1Array[1];

(continued)


c11.qxd 2/2/04 10:58 AM Page 160


//compute dominant cycle phasenDCPeriod = Math.floor( aV1Array[0] );nRealPart = 0.0;nImagPart = 0.0;

for ( x=0; x<nDCPeriod; x++ ) {nRealPart += Math.sin( DegToRad_

( 360*x/nDCPeriod ) ) * ( aCycleArray[x] );

nImagPart += Math.cos( DegToRad_( 360*x/nDCPeriod ) ) * ( aCycleArray[x] );

}

if ( Math.abs( nImagPart ) > 0.001 ) nDCPhase = RadToDeg( Math.atan_( nRealPart/nImagPart ) );

if ( Math.abs( nImagPart ) <= 0.001 ) nDCPhase = 90 * ( nRealPart<0 ? -1 : 1 );

nDCPhase += 90;if ( nImagPart < 0 ) nDCPhase += 180;

//return the calculated valuesif (!isNaN( nDCPhase ) ) {

return new Array( Math.sin( DegToRad_( nDCPhase) ), Math.sin( DegToRad_( nDCPhase+45 ) ) );

}

}

//== Convert Degrees to Radiansfunction DegToRad( nValue ) {var nTmp;

nTmp = nValue * ( Math.PI / 180 );return( nTmp );

}


c11.qxd 2/2/04 10:58 AM Page 161


//== Convert Radians to Degreesfunction RadToDeg( nValue ) {var nTmp;

nTmp = nValue * ( 180 / Math.PI );

return( nTmp );

}






}//if even # elements, take average of two_

middle elementselse {


}}

(continued)


c11.qxd 2/2/04 10:58 AM Page 162



}else {

return( 1 );}

}


Sinewave Indicator do not cross. In other words, the Sinewave Indicator indicates that some kind of trend-following system should be used. Then there are three clear cyclic turning points until the trend is indicated again in November. This is a case where the phase is unwinding and there is no clear cyclic crossover in the indicator. The Sinewave Indicator then has six successive sterling turning points identified until the trend returns at the right side of the chart, near the end of February.

FIGURE 11.4 The Sinewave Indicator Always Gives an Advanced Turning Point Warning

c11.qxd 2/2/04 10:58 AM Page 163


FIGURE 11.5 The Sinewave Indicator Gives Correct Cycle Signals


• The Sinewave Indicator is a noncausal predictive filter based on the premise that the Dominant Cycle has existed in the immediate past and will continue into the immediate future.

• The phase has a constant rate of change when the market is in a Cycle Mode.

• The phase languishes when the market is in a Trend Mode, and can even have a negative rate of change.

• The Sinewave Indicator consists of the sine of the Dominant Cycle phase and the sine of the Dominant Cycle phase advanced by 45°.

• The Sinewave Indicator gives entry and exit signals 1⁄16 of a cycle period in advance of the cycle turning point.

• The Sinewave Indicator seldom gives false whipsaw signals when the market is in a Trend Mode.

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c12.qxd 2/2/04 10:59 AM Page 165

CHAPTER 12

Adapting to the Trend

“I have no idea,” said Tom thoughtlessly.

A t this point, I have developed enough tools for you that you can now start putting them together to create some serious trading strate-gies. This chapter gives you the beginning of one such strategy. You

can use this strategy as a beginning and add your own rules to increase the percentage winners.

In previous chapters, I derived and adapted oscillator-type indicators with the goal of having the indicators move with the cycle component of the market with as little lag as possible. Most technical analysis trend-following techniques don’t use oscillators; they use moving averages or some variation thereof. In this chapter I will show you how to use the cycle measurement both as a trend indicator and as a trading system.

The slopes from any given point in a cycle to the same point in the next cycle are exactly the same. It doesn’t matter whether the point you select is the peak, the valley, or anyplace in between; the slope between the same points in idealized cycles is zero. If there is a difference in the amplitudes between successive samples, either the cycle period has changed or the mar-ket is in a trend. Since the cycle periods morph very slowly from cycle to cycle, it is more likely that the one-cycle momentum is an indication of the trend.

Our approach to forming this trading strategy is to measure the Dominant Cycle period and then use that measured period to take a one-cycle momentum. Momentum functions are notoriously noisy, so I smooth the momentum using the three-pole Super Smoother filter described in the next chapter. It is just that simple. The EasyLanguage and eSignal Formula Script (EFS) codes to compute the Smoothed Adaptive Momentum are shown in Figures 12.1 and 12.2, respectively.

165

c12.qxd 2/2/04 10:59 AM Page 166


{*****************************************************Smoothed Adaptive Momentum

*****************************************************}Inputs: Price((H+L)/2),

alpha(.07),Cutoff(8);

Vars: Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),Num(0),Denom(0),a1(0),b1(0),c1(0),coef1(0),coef2(0),coef3(0),coef4(0),Filt3(0);





I1 = Cycle[3];


FIGURE 12.1 EasyLanguage Code to Compute the Smoothed Adaptive Momentum

c12.qxd 2/2/04 10:59 AM Page 167





Value1 = Price - Price[IntPortion(Period - 1)];

a1 = expvalue(-3.14159 / Cutoff);b1 = 2*a1*Cosine(1.738*180 / Cutoff);c1 = a1*a1;coef2 = b1 + c1;coef3 = -(c1 + b1*c1);coef4 = c1*c1;coef1 = 1 - coef2 - coef3 - coef4;

Filt3 = coef1*Value1 + coef2*Filt3[1] + coef3*Filt3[2] + coef4*Filt3[3];

If CurrentBar < 4 then Filt3 = Value1;

Plot1(Filt3, “Filt3”);Plot2(0, “Ref”);


/*****************************************************Title: Smoothed Adaptive Momentum IndicatorCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

(continued)

FIGURE 12.2 EFS Code to Compute the Smoothed Adaptive Momentum

c12.qxd 2/2/04 10:59 AM Page 167






a1 = expvalue(-3.14159 / Cutoff);b1 = 2*a1*Cosine(1.738*180 / Cutoff);c1 = a1*a1;coef2 = b1 + c1;coef3 = -(c1 + b1*c1);coef4 = c1*c1;coef1 = 1 - coef2 - coef3 - coef4;



Plot1(Filt3, “Filt3”);Plot2(0, “Ref”);


/*****************************************************Title: Smoothed Adaptive Momentum IndicatorCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

(continued)

FIGURE 12.2 EFS Code to Compute the Smoothed Adaptive Momentum

c12.qxd 2/2/04 10:59 AM Page 168


======================================================Fix History:


======================================================*****************************************************/


var nBarCount = 0;

var aPriceArray = new Array(); var aSmoothArray = new Array(); var aCycleArray = new Array(); var aDeltaPhase = new Array(); var aPeriod = new Array(); var aInstPeriod = new Array(); var aQ1 = new Array(); var aI1 = new Array(); var aFiltArray = new Array();



setPriceStudy(false);setStudyTitle(“Smoothed Adaptive Momentum”);setCursorLabelName(“Filt3”, 0);setDefaultBarFgColor( Color.blue, 0 );addBand( 0.0, PS_SOLID, 1, Color.black, -10 );


aPriceArray[x] = 0.0;aSmoothArray[x] = 0.0;aCycleArray[x] = 0.0;aQ1[x] = 0.0;


c12.qxd 2/2/04 10:59 AM Page 169


aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0; aFiltArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha, Cutoff ) {var x;var nValue1;var nDC;var nOffset;var nCoef1;var nCoef2;var nCoef3;var nCoef4;var nA1;var nB1;var nC1;var nMedianDelta;


Alpha = 0.07;}if ( Cutoff == null ) {

Cutoff = 8;}


}


nBarCount++;(continued)


c12.qxd 2/2/04 10:59 AM Page 170










aFiltArray.pop();aFiltArray.unshift( 0 );

}





}else {


c12.qxd 2/2/04 10:59 AM Page 171



}




- aI1[1]/aQ1[1]) / (1 + aI1[0]*aI1[1]/(aQ1[0]*aQ1[1]));



= 1.1;



}else {




nOffset = Math.floor( aPeriod[0] )-1;(continued)


c12.qxd 2/2/04 10:59 AM Page 172


if ( nOffset < 0 ) nOffset = 0;

nValue1 = aPriceArray[0] - aPriceArray[ nOffset ];

nA1 = Math.exp( -3.14159 / Cutoff );nB1 = 2*nA1 * Math.cos( DegToRad( 1.738 * 180

/ Cutoff ) );nC1 = nA1 * nA1;

nCoef2 = nB1 + nC1;nCoef3 = -( nC1 + nB1 * nC1 );nCoef4 = nC1 * nC1;nCoef1 = 1 - nCoef2 - nCoef3 - nCoef4;

if ( nBarCount < 4 ) {aFiltArray[0] = nValue1;

}else {

aFiltArray[0] = nCoef1*nValue1 + nCoef2*aFiltArray[1] + nCoef3*aFiltArray[2] + nCoef4*aFiltArray[3];

}

//return the calculated valuesif (!isNaN( aFiltArray[0] ) ) {

return( aFiltArray[0] );}

}

//== Convert Degrees to Radiansfunction DegToRad( nValue ) {var nTmp;

nTmp = nValue * ( Math.PI / 180 );return( nTmp );


c12.qxd 2/2/04 10:59 AM Page 173


}

//== Convert Radians to Degreesfunction RadToDeg( nValue ) {var nTmp;

nTmp = nValue * ( 180 / Math.PI );

return( nTmp );

}







elementselse {


}}

(continued)


c12.qxd 2/2/04 10:59 AM Page 174



}else {

return( 1 );}

}


Figure 12.3 suggests that the uptrend starts when the indicator crosses up through the zero line and a downtrend starts when the indicator crosses down through the zero line.

I converted the Smoothed Adaptive Momentum Indicator to an auto-matic strategy by writing the trading rules to buy each time the filter crosses up through zero and to sell short each time the filter crosses down through zero. I also added a money management stop. This simple but ele-gant trend-following automatic trading strategy produced the results shown in Table 12.1. The EasyLanguage and EFS codes for the Smoothed Adaptive Momentum strategy are in Figures 12.4 and 12.5, respectively.

FIGURE 12.3 Smoothed Adaptive Momentum as a Trend Indicator

c12.qxd 2/2/04 10:59 AM Page 175


TABLE 12.1 Sample Trading Results Using the Smoothed Adaptive Momentum Trading Strategy

Number Percent Profit Future Net Profit of Trades Profitable Factor Max DD

EC (4/81–3/03) $112,112 196 40.3% 2.03 ($8,137) JY (9/81–3/03) $160,950 277 39.7% 2.01 ($13,450) SF (6/76–3/03) $157,337 523 38.8% 1.64 ($13,587)

{*****************************************************Smoothed Adaptive Momentum

*****************************************************}Inputs: Price((H+L)/2),

alpha(.07),Cutoff(8);

Vars: Smooth(0),Cycle(0),Q1(0),I1(0),DeltaPhase(0),MedianDelta(0),DC(0),InstPeriod(0),Period(0),Num(0),Denom(0),a1(0),b1(0),c1(0),coef1(0),coef2(0),coef3(0),coef4(0),Filt3(0);



(continued)

FIGURE 12.4 EasyLanguage Code for the Smoothed Adaptive Momentum Strategy

c12.qxd 2/2/04 10:59 AM Page 176




I1 = Cycle[3];






a1 = expvalue(-3.14159 / Cutoff);b1 = 2*a1*Cosine(1.738*180 / Cutoff);c1 = a1*a1;coef2 = b1 + c1;coef3 = -(c1 + b1*c1);coef4 = c1*c1;coef1 = 1 - coef2 - coef3 -coef4;



If Filt3 Crosses Over 0 then Buy Next Bar on Open;If Filt3 Crosses Under 0 then Sell Short Next Bar

on Open;


c12.qxd 2/2/04 10:59 AM Page 177


/*****************************************************Title: Smoothed Adaptive Momentum Trading

StrategyCoded By: Chris D. Kryza (Divergence Software, Inc.)Email: [email protected]: 07/09/2003Version: 1.0.0

======================================================Fix History:


======================================================******************************************************/


var nBarCount = 0;

var aPriceArrayvar aSmoothArrayvar aCycleArrayvar aDeltaPhasevar aPeriodvar aInstPeriodvar aQ1var aI1var aFiltArray

= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();= new Array();

var nStatus = 0; var nEntryPrice = 0; var nStop = 0; var nPVal = 0; var nSVal = 0;

var grID = 0;

(continued)

FIGURE 12.5 EFS for the Smoothed Adaptive Momentum Strategy

c12.qxd 2/2/04 10:59 AM Page 178




setPriceStudy( true );setStudyTitle(“Smoothed Adaptive Momentum Strategy”);

setShowCursorLabel( false );


aPriceArray[x] = 0.0; aSmoothArray[x] = 0.0; aCycleArray[x] = 0.0; aQ1[x] = 0.0; aI1[x] = 0.0; aDeltaPhase[x] = 0.0; aPeriod[x] = 0.0; aInstPeriod[x] = 0.0; aFiltArray[x] = 0.0;

}

}

//== Main processing functionfunction main( Alpha, Cutoff, StopAmt, PointValue ) {var x;var nValue1;var nDC;var nOffset;var nCoef1;var nCoef2;var nCoef3;var nCoef4;var nA1;var nB1;var nC1;var nMedianDelta;



c12.qxd 2/2/04 10:59 AM Page 179


Alpha = 0.07;}if ( Cutoff == null ) {

Cutoff = 8;}if ( StopAmt == null ) {

StopAmt = 1000.0;}if ( PointValue == null ) {

PointValue = 50;}

nSVal = StopAmt;nPVal = PointValue;


}


nBarCount++;







(continued)


Documents

Ehlers, John F. - Cybernetic Analysis for Stocks and Futures [Part 1]